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 Th24:
  f is one-to-one & g is one-to-one implies g*f is one-to-one
proof
  assume that
A1: f is one-to-one and
A2: g is one-to-one;
  now
    let x1,x2;
    assume
A3: x1 in dom(g*f) & x2 in dom(g*f);
    then
A4: (g*f).x1 = g.(f.x1) & (g*f).x2 = g .(f.x2) by Th12;
A5: x1 in dom f & x2 in dom f by A3,Th11;
    assume
A6: (g*f).x1 = (g*f).x2;
    f.x1 in dom g & f.x2 in dom g by A3,Th11;
    then f.x1 = f.x2 by A2,A4,A6;
    hence x1 = x2 by A1,A5;
  end;
  hence thesis;
end;
