reserve X,X1,X2,Y,Y1,Y2 for set, p,x,x1,x2,y,y1,y2,z,z1,z2 for object;
reserve f,g,g1,g2,h for Function,
  R,S for Relation;

theorem
  f is one-to-one implies f|X is one-to-one
proof
  assume
A1: f is one-to-one;
  let x1,x2;
  assume that
A2: x1 in dom(f|X) and
A3: x2 in dom(f|X);
  x1 in dom f /\ X by A2,RELAT_1:61;
  then
A4: x1 in dom f by XBOOLE_0:def 4;
  x2 in dom f /\ X by A3,RELAT_1:61;
  then
A5: x2 in dom f by XBOOLE_0:def 4;
  (f|X).x1 = f.x1 & (f|X).x2 = f.x2 by A2,A3,Th46;
  hence thesis by A1,A4,A5;
end;
