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
  (ex g st g*f = id dom f) implies f is one-to-one
proof
  given g such that
A1: g*f = id dom f;
  dom(g*f) = dom f by A1;
  then rng f c= dom g by Th15;
  hence thesis by A1,Th25;
end;
