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 Th25:
  g*f is one-to-one & rng f c= dom g implies f is one-to-one
proof
  assume that
A1: g*f is one-to-one and
A2: rng f c= dom g;
  now
    let x1,x2;
    assume that
A3: x1 in dom f & x2 in dom f and
A4: f.x1 =f.x2;
A5: x1 in dom(g*f) & x2 in dom(g*f) by A2,A3,RELAT_1:27;
    (g*f).x1 = g.(f.x1) & (g*f).x2 = g.(f.x2) by A3,Th13;
    hence x1 = x2 by A1,A4,A5;
  end;
  hence thesis;
end;
