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 Y|`f is one-to-one
proof
  assume
A1: f is one-to-one;
  let x1,x2 such that
A2: x1 in dom(Y|`f) & x2 in dom(Y|`f) and
A3: (Y|`f).x1 = (Y|`f).x2;
A4: x1 in dom f & x2 in dom f by A2,Th52;
  (Y|`f).x1 = f.x1 & (Y|`f).x2 = f.x2 by A2,Th52;
  hence thesis by A1,A3,A4;
end;
