I mentioned The family Way as an example of a theme composed by Paul without Lennon credits ... so the agreement in that case was somehow broken ... Lennon's solo project (Give Peace A Chance) was another example ...
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McCARTNEY. YESTERDAY AND TODAY (by Ray Coleman)
Chapter 5 - The Enigma of John Lennon
'He wore a suit of armour' "[...]The healthy partnership and camaraderie that evolved from Paul and John's competitive streak was only one step away from sibling rivalry. It now transpires that one of John's earliest 'hurts' inflicted by Paul was McCartney's solo writing of the music for the Hayley Mills film
The Family Way in 1966. 'I was told recently by Yoko that one of the things that hurt John over the years was me going off and doing The Family Way," Paul says. The filmmaking Boulting brothers had approached him via George Martin. 'I thought this was a great opportunity. We were all free to do stuff outside the Beatles and we'd each done various little things.'
When he mentioned it to John, Paul said, 'He would have had his suit of armour on and said: "No, I don't mind." However, my reasoning would be that at exactly the same time he went off to make a film. He wrote his books
[In His Own Write and A Spaniard In The Works]. It was in the spirit of all that. But what I didn't realize was that this was the first time one of us had done it on songs. John would write a book and I was supposed not to be jealous, which I wasn't. He acted in a film
[How I Won the War]. But I didn't realize he made a distinction between all those solo things and actually writing music because this was the first time one of us had done it in film scoring. I suppose what I should have said was: "I'd like to write it with John," and then that would have been OK. It actually didn't occur to me at that time at all. So I went off, saw and liked the film, said: "Right, come on George [Martin]," and I must say it was all over very quickly.' He was especially proud of the speed with which he wrote the song 'Love in the Open Air', 'which picked up an Ivor Novello award as the best film song that year, which I was always very proud of. [...]"
from here:http://wingspan.ru/bookseng/coleman/coleman05.html
P.S. Interesting reading ... Paul's view of John