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

assume A4:
for i being Element of I holds F . i is associative
; :: thesis: F is associative assume A3:
F is associative
; :: thesis: for i being Element of I holds F . i is associative

let i be Element of I; :: thesis: F . i is associative

ex Fi being non empty associative multMagma st Fi = F . i by A3;

hence F . i is associative ; :: thesis: verum

end;let i be Element of I; :: thesis: F . i is associative

ex Fi being non empty associative multMagma st Fi = F . i by A3;

hence F . i is associative ; :: thesis: verum

let i be set ; :: according to GROUP_7:def 4 :: thesis: ( i in I implies ex Fi being non empty associative multMagma st Fi = F . i )

assume i in I ; :: thesis: ex Fi being non empty associative multMagma st Fi = F . i

then reconsider i1 = i as Element of I ;

reconsider Fi = F . i1 as non empty associative multMagma by A4;

take Fi ; :: thesis: Fi = F . i

thus Fi = F . i ; :: thesis: verum