let X1, X2 be Subset of I; :: thesis: ( ( for i being object holds
( i in X1 iff ex G being Group st
( G = F . i & a . i <> 1_ G & i in I ) ) ) & ( for i being object holds
( i in X2 iff ex G being Group st
( G = F . i & a . i <> 1_ G & i in I ) ) ) implies X1 = X2 )
assume that
A2: for i being object holds
( i in X1 iff ex G being Group st
( G = F . i & a . i <> 1_ G & i in I ) ) and
A3: for i being object holds
( i in X2 iff ex G being Group st
( G = F . i & a . i <> 1_ G & i in I ) ) ; :: thesis: X1 = X2

now :: thesis: for i being object holds
( i in X1 iff i in X2 )

let i be object ; :: thesis: ( i in X1 iff i in X2 )
( i in X1 iff ex G being Group st
( G = F . i & a . i <> 1_ G & i in I ) ) by A2;
hence ( i in X1 iff i in X2 ) by A3; :: thesis: verum

end;

hence X1 = X2 by TARSKI:2; :: thesis: verum