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
  a is odd implies ex b st a|^2 + b|^2 = (b+1)|^2
  proof
    assume a is odd;
    then consider k be Nat such that
    A2: a = 2*k+1 by ABIAN:9;
    (2*k+1)|^2 + (2*k|^2 +2*k)|^2 = ((2*k|^2 +2*k) +1)|^2 by Th2;
    hence thesis by A2;
  end;
