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 Th18:
  R is antisymmetric implies R |_2 Y is antisymmetric
proof
  assume
A1: R is antisymmetric;
  now
    let a,b;
    assume [a,b] in R |_2 Y & [b,a] in R |_2 Y;
    then [a,b] in R & [b,a] in R by XBOOLE_0:def 4;
    hence a = b by A1,Lm3;
  end;
  hence thesis by Lm3;
end;
