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 Th74:
  R|(X \ Y) = R|X \ R|Y
proof
  let x,y;
  hereby
    assume
A1: [x,y] in R|(X \ Y); then
A2: x in X \ Y by Def9;
    then not x in Y by XBOOLE_0:def 5;
    then
A3: not [x,y] in R|Y by Def9;
    [x,y] in R by A1,Def9;
    then [x,y] in R|X by A2,Def9;
    hence [x,y] in R|X \ R|Y by A3,XBOOLE_0:def 5;
  end;
  assume
A4: [x,y] in R|X \ R|Y; then
A5: [x,y] in R by Def9;
    not [x,y] in R|Y by A4,XBOOLE_0:def 5; then
A6: not x in Y or not [x,y] in R by Def9;
    x in X by A4,Def9;
    then x in X \ Y by A4,A6,Def9,XBOOLE_0:def 5;
    hence [x,y] in R|(X \ Y) by A5,Def9;
end;
