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
  R"(X \/ Y) = R"X \/ R"Y
  proof
    thus R"(X \/ Y) c= R"X \/ R"Y
    proof
      let x be object;
      assume x in R"(X \/ Y);
      then consider y such that
A1:   [x,y] in R and
A2:   y in X \/ Y by Def12;
      y in X or y in Y by A2,XBOOLE_0:def 3;
      then x in R"X or x in R"Y by A1,Def12;
      hence x in R"X \/ R"Y by XBOOLE_0:def 3;
    end;
    let x be object;
    assume
A3: x in R"X \/ R"Y;
    per cases by A3,XBOOLE_0:def 3;
    suppose x in R"Y;
      then consider y such that
A4:   [x,y] in R and
A5:   y in Y by Def12;
      y in X \/ Y by A5,XBOOLE_0:def 3;
      hence x in R"(X \/ Y) by A4,Def12;
    end;
    suppose x in R"X;
      then consider y such that
A6:   [x,y] in R and
A7:   y in X by Def12;
      y in X \/ Y by A7,XBOOLE_0:def 3;
      hence x in R"(X \/ Y) by A6,Def12;
    end;
  end;
