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 Th15:
  <%a,b%> is left_complementable implies
  a is left_complementable & b is left_complementable
  proof
    set C = Cayley-Dickson(N);
    given x being Element of C such that
A1: x+<%a,b%> = 0.C;
    consider x1,x2 being Element of N such that
A2: x = <%x1,x2%> by Th12;
A3: 0.C = <%0.N,0.N%> by Def9;
A4: <%x1,x2%>+<%a,b%> = <%x1+a,x2+b%> by Def9;
    hereby
      take x1;
      thus x1+a = 0.N by A1,A2,A3,A4,Th3;
    end;
    take x2;
    thus x2+b = 0.N by A1,A2,A3,A4,Th3;
  end;
