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 \ R"Y c= R"(X \ Y)
proof
  let x be object;
  assume
A1: x in R"X \ R"Y;
  then consider y such that
A2: [x,y] in R and
A3: y in X by Def12;
  not x in R"Y by A1,XBOOLE_0:def 5;
  then not y in Y or not [x,y] in R by Def12;
  then y in X \ Y by A2,A3,XBOOLE_0:def 5;
  hence thesis by A2,Def12;
end;
