theorem
  Half 0.G = 0.G & Half (x+y) = Half x + Half y & (Half x = Half y
  implies x = y) & Half Double x = x
proof
  Double 0.G = 0.G by RLVECT_1:def 4;
  hence Half 0.G = 0.G by Def6;
  Double (Half x + Half y) = Double Half x + Double Half y by Th10
    .= x + Double Half y by Def6
    .= x + y by Def6;
  hence Half (x+y) = Half x + Half y by Def6;
  thus Half x = Half y implies x = y
  proof
    assume Half x = Half y;
    hence x = Double Half y by Def6
      .= y by Def6;
  end;
  thus thesis by Def6;
end;
