let A, B be set ; :: thesis: ( ( for x being object holds
( x in A iff ex K being strict Field st
( K = x & F is Subfield of K & K is Subfield of E ) ) ) & ( for x being object holds
( x in B iff ex K being strict Field st
( K = x & F is Subfield of K & K is Subfield of E ) ) ) implies A = B )
assume that
A3: for x being object holds
( x in A iff ex K being strict Field st
( K = x & F is Subfield of K & K is Subfield of E ) ) and
A4: for x being object holds
( x in B iff ex K being strict Field st
( K = x & F is Subfield of K & K is Subfield of E ) ) ; :: thesis: A = B

A5: now :: thesis: for x being object st x in A holds
x in B

let x be object ; :: thesis: ( x in A implies x in B )
assume x in A ; :: thesis: x in B
then consider K being strict Field such that
A6: ( K = x & F is Subfield of K & K is Subfield of E ) by A3;
thus x in B by A6, A4; :: thesis: verum

end;

now :: thesis: for x being object st x in B holds
x in A

let x be object ; :: thesis: ( x in B implies x in A )
assume x in B ; :: thesis: x in A
then consider K being strict Field such that
A6: ( K = x & F is Subfield of K & K is Subfield of E ) by A4;
thus x in A by A6, A3; :: thesis: verum

end;

hence A = B by A5, TARSKI:2; :: thesis: verum