reserve A,X,X1,X2,Y,Y1,Y2 for set, a,b,c,d,x,y,z for object;
reserve P,P1,P2,Q,R,S for Relation;

theorem Th72:
  R|(X \/ Y) = (R|X) \/ (R|Y)
proof
  let x,y;
  hereby
    assume
A1: [x,y] in R|(X \/ Y);
    then x in X \/ Y by Def9; then
A2: x in X or x in Y by XBOOLE_0:def 3;
    [x,y] in R by A1,Def9;
    then [x,y] in R|X or [x,y] in R|Y by A2,Def9;
    hence [x,y] in (R|X) \/ (R|Y) by XBOOLE_0:def 3;
  end;
  assume
A3: [x,y] in (R|X) \/ (R|Y);
  per cases by A3,XBOOLE_0:def 3;
  suppose
A4: [x,y] in R|Y;
    then x in Y by Def9;
    then
A5: x in X \/ Y by XBOOLE_0:def 3;
    [x,y] in R by A4,Def9;
    hence [x,y] in R|(X \/ Y) by A5,Def9;
  end;
  suppose
A6: [x,y] in R|X;
    then x in X by Def9;
    then
A7: x in X \/ Y by XBOOLE_0:def 3;
    [x,y] in R by A6,Def9;
    hence [x,y] in R|(X \/ Y) by A7,Def9;
  end;
end;
