let Y be non empty set ; :: thesis:
let P, Q, R be a_partition of Y; :: thesis:
assume A1: ERl R = (ERl P) * (ERl Q) ; :: thesis:
let x, y be Element of Y; :: thesis:

hereby :: thesis:

assume x in EqClass y,R ; :: thesis:
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 A1, RELSET_1:51;
take z = z; :: thesis:
thus x in EqClass z,P by A2, Th5; :: thesis:
thus z in EqClass y,Q by A3, Th5; :: thesis:

end;

given z being Element of Y such that A4: x in EqClass z,P and
A5: z in EqClass y,Q ; :: thesis:
A6: [x,z] in ERl P by A4, Th5;
[z,y] in ERl Q by A5, Th5;
then [x,y] in ERl R by A1, A6, RELAT_1:def 8;
hence x in EqClass y,R by Th5; :: thesis: