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 Th44:
  q < 0 & b < c & a|^2 +b|^2 = c|^2 implies
    a|^2 + (b+q)|^2 > (c+q)|^2
  proof assume
    A1: q < 0 & b < c & a|^2 +b|^2 = c|^2; then
    A1a: b - c < c - c by XREAL_1:9;
    a|^2 + (b+q)|^2 - (c+q)|^2 = 2*q*(b-c) by A1,Lm61; then
    a|^2 + (b+q)|^2 - (c+q)|^2 + (c+q)|^2 > 0+ (c+q)|^2 by A1,A1a,XREAL_1:6;
    hence thesis;
  end;
