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 Th43:
   |.a1*b2-a2*b1.|<>0 & q <> q1 &
   a2*denominator(q)+b2*numerator(q)=0 implies
   a2*denominator(q1)+b2*numerator(q1)<>0
   proof
     assume that
A1:  |.a1*b2-a2*b1.|<>0 and
A2:  q <> q1 and
A3:  a2*denominator(q)+b2*numerator(q)=0;
A5:  (-a2)*denominator(q)= numerator(q)*b2 by A3;
     per cases by A1;
       suppose
A4:     a2 <> 0 & b2 <> 0;
        assume a2*denominator(q1)+b2*numerator(q1) = 0; then
A7:     (-a2)*denominator(q1) = b2*numerator(q1);
        q = numerator(q)/denominator(q) by RAT_1:15
         .= (-a2)/b2 by A4,A5,XCMPLX_1:94
         .= numerator(q1)/denominator(q1) by A4,A7,XCMPLX_1:94
         .= q1 by RAT_1:15;
         hence contradiction by A2;
        end;
        suppose a2 <> 0 & b2 = 0;
          hence thesis;
        end;
        suppose
A10:      a2 = 0 & b2 <> 0;
          assume
A11:      a2*denominator(q1)+b2*numerator(q1) = 0;
          q = 0 by A3,A10 .= q1 by A10,A11;
          hence contradiction by A2;
         end;
       end;
