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 Th15:
  R is reflexive implies R |_2 X is reflexive
proof
  assume
A1: R is reflexive;
  now
    let a;
    assume
A2: a in field(R |_2 X);
    then a in X by Th12;
    then
A3: [a,a] in [:X,X:] by ZFMISC_1:87;
    a in field R by A2,Th12;
    then [a,a] in R by A1,Lm1;
    hence [a,a] in R |_2 X by A3,XBOOLE_0:def 4;
  end;
  hence thesis by Lm1;
end;
