reserve r1,r2,r3 for non negative Real;
reserve n,m1 for Nat;
reserve s for Real;
reserve cn,cd,i1,j1 for Integer;
reserve r for irrational Real;
reserve q for Rational;
reserve c0,c1,c2,u,a0,b0 for Real;
reserve a,b for Real;
reserve n for Integer;
reserve a1,a2,b1,b2,c1,c2 for Element of REAL;
reserve eps for positive Real;
reserve r1 for non negative Real;
reserve q,q1 for Element of RAT;

theorem Th45:
  for a1,b1 be Real, n1,d1 be Integer st d1 > 0 &
   |.a1/b1+n1/d1.|<1/(sqrt 5 * d1|^2)
   holds ex d be Real st n1/d1 = -a1/b1 + d/d1|^2 & |.d.| < 1/sqrt 5
   proof
     let a1,b1 be Real, n1,d1 be Integer;
     assume that
A1:  d1 > 0 and
A2:  |.a1/b1+n1/d1.|<1/(sqrt 5 * d1|^2);
     set d = (a1/b1+n1/d1)*d1|^2;
     d/d1|^2 = (a1/b1+n1/d1) by A1,XCMPLX_1:89; then
A4:  -a1/b1 + d/d1|^2 = n1/d1;
A5:  |.d .| = |.(a1/b1+n1/d1).|*|.d1|^2.| by COMPLEX1:65
   .= |.(a1/b1+n1/d1).|*(d1|^2);
     (1/(sqrt 5 * d1|^2))*(d1|^2)
    =(1/(sqrt 5))*(1/(d1|^2))*(d1|^2) by XCMPLX_1:102
   .=1/(sqrt 5) by A1,XCMPLX_1:109;
     hence thesis by A4,A1,A2,XREAL_1:68,A5;
   end;
