reserve a,b,c,d,x,j,k,l,m,n for Nat,
  p,q,t,z,u,v for Integer,
  a1,b1,c1,d1 for Complex;

theorem Th34:
  a+b divides a|^(2*m+1) + b|^(2*m+1)
  proof
    A2: a|^(2*m+2) - b|^(2*m+2)
    = (a|^(2*m+1)+b|^(2*m+1))*(a-b) + a*b*(a|^(2*m)-b|^(2*m)) by Th21;
    set n=m+1;
    consider t be Integer such that
    A4: a|^(2*m)-b|^(2*m) = (a|^2-b|^2)*t by Th33,INT_1:def 3;
    consider z be Integer such that
    A4a: a|^(2*n)-b|^(2*n) = (a|^2-b|^2)*z by Th33,INT_1:def 3;
    A5: (a-b)*(a|^(2*m+1)+b|^(2*m+1))
    = a|^(2*m+2) - b|^(2*m+2) - a*b*(a|^(2*m)-b|^(2*m)) by A2
    .= a|^(2*n) - b|^(2*n) - a*b*((a|^2-b|^2)*t) by A4
    .= (a|^2-b|^2)*(z -a*b*t) by A4a
    .= (a-b)*(a+b)*(z -a*b*t) by Th1
    .= (a-b)*((a+b)*(z -a*b*t));
X:  2*a = a+a;
    a|^(2*m+1)+b|^(2*m+1) = (a+b)*(z-a*b*t) or a-b = 0 by A5,XCMPLX_1:5;
    hence thesis by Lm46,X;
  end;
