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

theorem PT1:
  for a be non zero square Integer, b be Integer holds
  a*b is square implies b is square
  proof
  let a be non zero square Integer, b be Integer;
  assume
  A1: a*b is square;
  consider x be Nat such that
  A2: a = x^2 by PYTHTRIP:def 3;
  consider y be Nat such that
  A3: a*b = y^2 by A1,PYTHTRIP:def 3;
  x^2 divides y^2 by A2,A3; then
  |.x.| divides |.y.| by PYTHTRIP:6; then
  consider k be Integer such that
  A4: y = k*x;
  A5: k*k = k^2 & x*x = x^2 & y*y = y^2 by SQUARE_1:def 1;
  a*b = a*k^2 by A2,A3,A4,A5; then
  b = k^2 by XCMPLX_1:5
  .= |.k.|^2 by COMPLEX1:75;
  hence thesis;
end;
