let D1, D2 be LINE_DOMAIN of V; :: thesis: ( ( for x being object holds

( x in D1 iff x is LINE of V ) ) & ( for x being object holds

( x in D2 iff x is LINE of V ) ) implies D1 = D2 )

assume that

A3: for x being object holds

( x in D1 iff x is LINE of V ) and

A4: for x being object holds

( x in D2 iff x is LINE of V ) ; :: thesis: D1 = D2

( x in D1 iff x is LINE of V ) ) & ( for x being object holds

( x in D2 iff x is LINE of V ) ) implies D1 = D2 )

assume that

A3: for x being object holds

( x in D1 iff x is LINE of V ) and

A4: for x being object holds

( x in D2 iff x is LINE of V ) ; :: thesis: D1 = D2

now :: thesis: for x being object holds

( x in D1 iff x in D2 )

hence
D1 = D2
by TARSKI:2; :: thesis: verum( x in D1 iff x in D2 )

let x be object ; :: thesis: ( x in D1 iff x in D2 )

( x in D1 iff x is LINE of V ) by A3;

hence ( x in D1 iff x in D2 ) by A4; :: thesis: verum

end;( x in D1 iff x is LINE of V ) by A3;

hence ( x in D1 iff x in D2 ) by A4; :: thesis: verum