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

assume A1: F is Group-like ; :: thesis: for i being Element of I holds F . i is Group-like
let i be Element of I; :: thesis: F . i is Group-like
ex Fi being non empty Group-like multMagma st Fi = F . i by A1;
hence F . i is Group-like ; :: thesis: verum

end;

assume A2: for i being Element of I holds F . i is Group-like ; :: thesis: F is Group-like
let i be set ; :: according to GROUP_7:def 3 :: thesis: ( i in I implies ex Fi being non empty Group-like multMagma st Fi = F . i )
assume i in I ; :: thesis: ex Fi being non empty Group-like multMagma st Fi = F . i
then reconsider i1 = i as Element of I ;
reconsider F1 = F . i1 as non empty Group-like multMagma by A2;
take F1 ; :: thesis: F1 = F . i
thus F1 = F . i ; :: thesis: verum