let A, B be set ; :: thesis: ( ( for o being object holds
( o in A iff ex K being strict Field st
( o = K & K is Subfield of F ) ) ) & ( for o being object holds
( o in B iff ex K being strict Field st
( o = K & K is Subfield of F ) ) ) implies A = B )
assume that
A1: for x being object holds
( x in A iff ex K being strict Field st
( x = K & K is Subfield of F ) ) and
A2: for x being object holds
( x in B iff ex K being strict Field st
( x = K & K is Subfield of F ) ) ; :: thesis: A = B

A3: 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 ex K being strict Field st
( x = K & K is Subfield of F ) by A1;
hence x in B by A2; :: 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 ex K being strict Field st
( x = K & K is Subfield of F ) by A2;
hence x in A by A1; :: thesis: verum

end;

hence A = B by A3, TARSKI:2; :: thesis: verum