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 Th20:
  a is left_add-cancelable & b is left_add-cancelable implies
  <%a,b%> is left_add-cancelable
  proof
    assume
A1: a is left_add-cancelable & b is left_add-cancelable;
    let y,z be Element of Cayley-Dickson(N) such that
A2: <%a,b%>+y = <%a,b%>+z;
    consider y1,y2 being Element of N such that
A3: y = <%y1,y2%> by Th12;
    consider z1,z2 being Element of N such that
A4: z = <%z1,z2%> by Th12;
    <%a,b%>+<%y1,y2%> = <%a+y1,b+y2%> & <%a,b%>+<%z1,z2%> = <%a+z1,b+z2%>
    by Def9;
    then a+y1 = a+z1 & b+y2 = b+z2 by A2,A3,A4,Th3;
    then y1 = z1 & y2 = z2 by A1;
    hence y = z by A3,A4;
  end;
