hereby :: thesis: ( ( for i being Element of I holds F . i is commutative ) implies F is commutative )

assume A5: F is commutative ; :: thesis: for i being Element of I holds F . i is commutative
let i be Element of I; :: thesis: F . i is commutative
ex Fi being non empty commutative multMagma st Fi = F . i by A5;
hence F . i is commutative ; :: thesis: verum

end;

assume A6: for i being Element of I holds F . i is commutative ; :: thesis: F is commutative
let i be set ; :: according to GROUP_7:def 5 :: thesis: ( i in I implies ex Fi being non empty commutative multMagma st Fi = F . i )
assume i in I ; :: thesis: ex Fi being non empty commutative multMagma st Fi = F . i
then reconsider i1 = i as Element of I ;
reconsider Fi = F . i1 as non empty commutative multMagma by A6;
take Fi ; :: thesis: Fi = F . i
thus Fi = F . i ; :: thesis: verum