reserve V for non empty RLSStruct;
reserve x,y,y1 for set;
reserve v for VECTOR of V;
reserve a,b for Real;

theorem Th8:
  for V being add-associative right_zeroed right_complementable
  non empty addLoopStr, w,u,v1,v2 being Element of V st
  w + v1 = w + v2 or v1 + w = v2 + w holds v1 = v2
proof
  let V be add-associative right_zeroed right_complementable non empty
  addLoopStr;
  let w,u,v1,v2 be Element of V;
A1: now
    assume
A2: v1 + w = v2 + w;
    thus v1 = v1 + 0.V
      .= v1 + (w + -w) by Th5
      .= v1 + w + -w by Def3
      .= v2 + (w + -w) by A2,Def3
      .= v2 + 0.V by Th5
      .= v2;
  end;
  now
    assume
A3: w + v1 = w + v2;
    thus v1 = 0.V + v1
      .= -w + w + v1 by Th5
      .= -w + (w + v1) by Def3
      .= -w + w + v2 by A3,Def3
      .= 0.V + v2 by Th5
      .= v2;
  end;
  hence thesis by A1;
end;
