let X1, X2 be set ; :: thesis: ( ( for b being object holds
( b in X1 iff ex x being bound_QC-variable of Al st
( x = b & x in still_not-bound_in p & x = (@ Sub) . x ) ) ) & ( for b being object holds
( b in X2 iff ex x being bound_QC-variable of Al st
( x = b & x in still_not-bound_in p & x = (@ Sub) . x ) ) ) implies X1 = X2 )
assume that
A2: for b being object holds
( b in X1 iff ex x being bound_QC-variable of Al st
( x = b & x in still_not-bound_in p & x = (@ Sub) . x ) ) and
A3: for b being object holds
( b in X2 iff ex x being bound_QC-variable of Al st
( x = b & x in still_not-bound_in p & x = (@ Sub) . x ) ) ; :: thesis: X1 = X2

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

let b be object ; :: thesis: ( b in X1 iff b in X2 )
( b in X1 iff ex x being bound_QC-variable of Al st
( x = b & x in still_not-bound_in p & x = (@ Sub) . x ) ) by A2;
hence ( b in X1 iff b in X2 ) by A3; :: thesis: verum

end;

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