theorem Th71:
  f is dilatation & g is dilatation implies (f*g) is dilatation
proof
  assume
A1: f is dilatation & g is dilatation;
  now
    let x,y;
    set x9=g.x;
    set y9=g.y;
A2: (f*g).x= f.x9 & (f*g).y=f.y9 by FUNCT_2:15;
A3: now
      assume x9=y9;
      then x=y by FUNCT_2:58;
      hence x,y // (f*g).x,(f*g).y by AFF_1:3;
    end;
    x,y // x9,y9 & x9,y9 // f.x9,f.y9 by A1,Th68;
    hence x,y // (f*g).x,(f*g).y by A2,A3,AFF_1:5;
  end;
  hence thesis by Th68;
end;
