theorem Th72:
  f is dilatation implies for a,b,c,d holds a,b // c,d iff f.a,f.b // f.c,f.d
proof
  assume
A1: f is dilatation;
  let a,b,c,d;
A2: c,d // f.c,f.d by A1,Th68;
A3: a,b // f.a,f.b by A1,Th68;
A4: now
A5: now
A6:   now
        assume f.c = f.d;
        then c = d by FUNCT_2:58;
        hence a,b // c,d by AFF_1:3;
      end;
      assume a,b // f.c,f.d;
      hence a,b // c,d by A2,A6,AFF_1:5;
    end;
A7: now
      assume f.a=f.b;
      then a=b by FUNCT_2:58;
      hence a,b // c,d by AFF_1:3;
    end;
    assume f.a,f.b // f.c,f.d;
    hence a,b // c,d by A3,A7,A5,AFF_1:5;
  end;
  now
A8: now
A9:   c = d implies f.a,f.b // f.c,f.d by AFF_1:3;
      assume f.a,f.b // c,d;
      hence f.a,f.b // f.c,f.d by A2,A9,AFF_1:5;
    end;
    assume a,b // c,d;
    then f.a,f.b // c,d or a=b by A3,AFF_1:5;
    hence f.a,f.b // f.c,f.d by A8,AFF_1:3;
  end;
  hence thesis by A4;
end;
