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 Th27:
  N is add-associative right_zeroed right_complementable implies
  - <% a,b %> = <% -a,-b %>
  proof
    assume
A1: N is add-associative right_zeroed right_complementable;
    then
A2: a+-a = 0.N & b+-b = 0.N by RLVECT_1:def 10;
    <%a,b%> + <%-a,-b%> = <%a+-a,b+-b%> by Def9
    .= 0.Cayley-Dickson(N) by A2,Def9;
    hence thesis by A1,RLVECT_1:def 10;
  end;
