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 Th21:
  <%a,b%> is left_add-cancelable implies
  a is left_add-cancelable & b is left_add-cancelable
  proof
    assume
A1: <%a,b%> is left_add-cancelable;
    hereby
      let y,z be Element of N such that
A2:   a+y = a+z;
      <%a,b%>+<%y,0.N%> = <%a+y,b+0.N%> by Def9
      .= <%a,b%>+<%z,0.N%> by A2,Def9;
      then <%y,0.N%> = <%z,0.N%> by A1;
      hence y = z by Th3;
    end;
    let y,z be Element of N such that
A3: b+y = b+z;
    <%a,b%>+<%0.N,y%> = <%a+0.N,b+y%> by Def9
    .= <%a,b%>+<%0.N,z%> by A3,Def9;
    then <%0.N,y%> = <%0.N,z%> by A1;
    hence y = z by Th3;
  end;
