let Y be non empty set ; :: thesis: for P, Q, R being a_partition of Y st ERl R = (ERl P) * (ERl Q) holds
for x, y being Element of Y holds
( x in EqClass (y,R) iff ex z being Element of Y st
( x in EqClass (z,P) & z in EqClass (y,Q) ) )

let P, Q, R be a_partition of Y; :: thesis: ( ERl R = (ERl P) * (ERl Q) implies for x, y being Element of Y holds
( x in EqClass (y,R) iff ex z being Element of Y st
( x in EqClass (z,P) & z in EqClass (y,Q) ) ) )

assume A1: ERl R = (ERl P) * (ERl Q) ; :: thesis: for x, y being Element of Y holds
( x in EqClass (y,R) iff ex z being Element of Y st
( x in EqClass (z,P) & z in EqClass (y,Q) ) )

let x, y be Element of Y; :: thesis: ( x in EqClass (y,R) iff ex z being Element of Y st
( x in EqClass (z,P) & z in EqClass (y,Q) ) )

hereby :: thesis: ( ex z being Element of Y st
( x in EqClass (z,P) & z in EqClass (y,Q) ) implies x in EqClass (y,R) )
assume x in EqClass (y,R) ; :: thesis: ex z being Element of Y st
( x in EqClass (z,P) & z in EqClass (y,Q) )

then [x,y] in ERl R by Th5;
then consider z being Element of Y such that
A2: [x,z] in ERl P and
A3: [z,y] in ERl Q by ;
take z = z; :: thesis: ( x in EqClass (z,P) & z in EqClass (y,Q) )
thus x in EqClass (z,P) by ; :: thesis: z in EqClass (y,Q)
thus z in EqClass (y,Q) by ; :: thesis: verum
end;
given z being Element of Y such that A4: ( x in EqClass (z,P) & z in EqClass (y,Q) ) ; :: thesis: x in EqClass (y,R)
( [x,z] in ERl P & [z,y] in ERl Q ) by ;
then [x,y] in ERl R by ;
hence x in EqClass (y,R) by Th5; :: thesis: verum