let D1, D2 be set ; :: thesis: ( ( for o being object holds
( o in D1 iff ex lamb being object st
( lamb in Lamb & o in divs (lamb,x,X,Inv) ) ) ) & ( for o being object holds
( o in D2 iff ex lamb being object st
( lamb in Lamb & o in divs (lamb,x,X,Inv) ) ) ) implies D1 = D2 )
assume that
A3: for o being object holds
( o in D1 iff ex lamb being object st
( lamb in Lamb & o in divs (lamb,x,X,Inv) ) ) and
A4: for o being object holds
( o in D2 iff ex lamb being object st
( lamb in Lamb & o in divs (lamb,x,X,Inv) ) ) ; :: thesis: D1 = D2

now :: thesis: for o being object holds
( o in D1 iff o in D2 )

let o be object ; :: thesis: ( o in D1 iff o in D2 )
( o in D1 iff ex lamb being object st
( lamb in Lamb & o in divs (lamb,x,X,Inv) ) ) by A3;
hence ( o in D1 iff o in D2 ) by A4; :: thesis: verum

end;

hence D1 = D2 by TARSKI:2; :: thesis: verum