reserve u,v,x,y,z,X,Y for set;
reserve r,s for Real;
reserve N for non empty ConjNormAlgStr;
reserve a,a1,a2,b,b1,b2 for Element of N;
reserve c,c1,c2 for Element of Cayley-Dickson(N);

theorem
  <% a,b %> is left_complementable right_add-cancelable implies
  - <% a,b %> = <% -a,-b %>
  proof
    assume
A1: <%a,b%> is left_complementable right_add-cancelable;
    then a is left_complementable & b is left_complementable &
    a is right_add-cancelable & b is right_add-cancelable by Th15,Th23;
    then
A2: -a+a = 0.N & -b+b = 0.N by ALGSTR_0:def 13;
    <%-a,-b%> + <%a,b%> = <%-a+a,-b+b%> by Def9
    .= 0.Cayley-Dickson(N) by A2,Def9;
    hence thesis by A1,ALGSTR_0:def 13;
  end;
