reserve r,s,t,x9,y9,z9,p,q for Element of RAT+;
reserve x,y,z for Element of REAL+;

theorem Th11:
  x + y = x + z implies y = z
proof
  assume
A1: x + y = x + z;
  consider q being Element of REAL+ such that
A2: z + q = y or y + q = z by Th10;
  per cases by A2;
  suppose
A3: z + q = y;
    then x + y = x + y + q by A1,Th6;
    then q = {} by Lm35;
    hence thesis by A3,Def8;
  end;
  suppose
A4: y + q = z;
    then x + z = x + z + q by A1,Th6;
    then q = {} by Lm35;
    hence thesis by A4,Def8;
  end;
end;
