 reserve x,y,z,t for object,X,Y,Z,W for set;
 reserve R,S,T for Relation;

theorem
  R c= [:X,Y:] & X = Z \/ W implies R = (R|Z) \/ (R|W)
proof
  assume that
A1: R c= [:X,Y:] and
A2: X = Z \/ W;
  thus R = R /\ [:X,Y:] by A1,XBOOLE_1:28
    .= R /\ ([:Z,Y:] \/ [:W,Y:]) by A2,ZFMISC_1:97
    .= (R /\ [:Z,Y:]) \/ (R /\ [:W,Y:]) by XBOOLE_1:23
    .= (R|Z) \/ (R /\ [:W,Y:]) by A1,Th8
    .= (R|Z) \/ (R|W) by A1,Th8;
end;
