theorem Th15:
  W.(a,p) = x iff (a,x).W = p
proof
  thus W.(a,p) = x implies (a,x).W = p
  proof
    assume
A1: W.(a,p) = x;
    now
      let m;
      W.(a,p.m) = x.m by A1,Def9;
      hence (a,x.m).W = p.m by MIDSP_2:33;
    end;
    hence thesis by Def8;
  end;
  thus (a,x).W = p implies W.(a,p) = x
  proof
    assume
A2: (a,x).W = p;
    now
      let m;
      (a,x.m).W = p.m by A2,Def8;
      hence W.(a,p.m) = x.m by MIDSP_2:33;
    end;
    hence thesis by Def9;
  end;
end;
