let S1, S2 be set ; :: thesis: ( ( for x being object holds
( x in S1 iff ex W being strict free Submodule of V st
( W = x & rank W = n ) ) ) & ( for x being object holds
( x in S2 iff ex W being strict free Submodule of V st
( W = x & rank W = n ) ) ) implies S1 = S2 )

assume that
A6: for x being object holds
( x in S1 iff ex W being strict free Submodule of V st
( W = x & rank W = n ) ) and
A7: for x being object holds
( x in S2 iff ex W being strict free Submodule of V st
( W = x & rank W = n ) ) ; :: thesis: S1 = S2
for x being object holds
( x in S1 iff x in S2 )
proof
let x be object ; :: thesis: ( x in S1 iff x in S2 )
hereby :: thesis: ( x in S2 implies x in S1 )
assume x in S1 ; :: thesis: x in S2
then ex W being strict free Submodule of V st
( W = x & rank W = n ) by A6;
hence x in S2 by A7; :: thesis: verum
end;
assume x in S2 ; :: thesis: x in S1
then ex W being strict free Submodule of V st
( W = x & rank W = n ) by A7;
hence x in S1 by A6; :: thesis: verum
end;
hence S1 = S2 by TARSKI:2; :: thesis: verum