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 Th44:
   for a1,a2,b1,b2,r1 st |.a1*b2-a2*b1.|<>0 holds
   ex q be Element of RAT st denominator(q) > [\r1/]+1 & q in HWZSet(r) &
   a2*denominator(q)+b2*numerator(q) <> 0
   proof
     let a1,a2,b1,b2,r1;
     assume
A1:  |.a1*b2-a2*b1.|<>0;
     consider q be Element of RAT such that
A2:  denominator(q) > [\r1/]+1 and
A3:  q in HWZSet(r) by Th42;
     per cases;
       suppose a2*denominator(q)+b2*numerator(q)<>0;
         hence thesis by A2,A3;
       end;
       suppose
A5:      a2*denominator(q)+b2*numerator(q)=0;
         consider q1 be Element of RAT such that
A6:      denominator(q1) > [\ denominator(q) /]+1 and
A7:      q1 in HWZSet(r) by Th42;
         denominator(q1) > denominator(q) by A6,INT_1:29,XXREAL_0:2; then
A8:      denominator(q1) > [\r1/]+1 by A2,XXREAL_0:2;
         q1 <> q by A6,INT_1:29; then
         a2*denominator(q1)+b2*numerator(q1)<>0 by A1,A5,Th43;
         hence thesis by A7,A8;
       end;
     end;
