reserve a,b,c,d,x,y,z for object, X,Y,Z for set;
reserve R,S,T for Relation;
reserve F,G for Function;

theorem Th10:
  R |_2 X = X|`R|X
proof
  let x,y be object;
  thus [x,y] in R |_2 X implies [x,y] in X|`R|X
  proof
    assume
A1: [x,y] in R |_2 X;
    then
A2: [x,y] in [:X,X:] by XBOOLE_0:def 4;
    then
A3: x in X by ZFMISC_1:87;
A4: y in X by A2,ZFMISC_1:87;
    [x,y] in R by A1,XBOOLE_0:def 4;
    then [x,y] in X|`R by A4,RELAT_1:def 12;
    hence thesis by A3,RELAT_1:def 11;
  end;
  assume
A5: [x,y] in X|`R|X;
  then
A6: [x,y] in X|`R by RELAT_1:def 11;
  then
A7: [x,y] in R by RELAT_1:def 12;
A8: y in X by A6,RELAT_1:def 12;
  x in X by A5,RELAT_1:def 11;
  then [x,y] in [:X,X:] by A8,ZFMISC_1:87;
  hence thesis by A7,XBOOLE_0:def 4;
end;
