let f1, f2 be Homomorphism-Family of H, FreeProduct H; :: thesis: ( ( for i being Element of I holds f1 . i = injection (H,i) ) & ( for i being Element of I holds f2 . i = injection (H,i) ) implies f1 = f2 )
assume that
A2: for i being Element of I holds f1 . i = injection (H,i) and
A3: for i being Element of I holds f2 . i = injection (H,i) ; :: thesis: f1 = f2

now :: thesis: for x being object st x in I holds
f1 . x = f2 . x

let x be object ; :: thesis: ( x in I implies f1 . x = f2 . x )
assume x in I ; :: thesis: f1 . x = f2 . x
then reconsider i = x as Element of I ;
thus f1 . x = injection (H,i) by A2
.= f2 . x by A3 ; :: thesis: verum

end;

hence f1 = f2 by PBOOLE:3; :: thesis: verum