reserve i,j,k,n,m,l,s,t for Nat;
reserve a,b for Real;
reserve F for real-valued FinSequence;
reserve z for Complex;
reserve x,y for Complex;
reserve r,s,t for natural Number;

theorem Th39:
  m<>1 & m divides n implies not m divides (n+1)
proof
  assume that
A1: m<>1 and
A2: m divides n and
A3: m divides (n+1);
  consider t being Nat such that
A4: n = m * t by A2,NAT_D:def 3;
  consider s being Nat such that
A5: n+1 = m * s by A3,NAT_D:def 3;
  t <= s
  proof
    (n+1)*t=m*s*t by A5
      .= n*s by A4;
    then
A6: t=n*s/(n+1) by XCMPLX_1:89
      .=s*(n/(n+1)) by XCMPLX_1:74;
    assume
A7: t > s;
    s>0 by A5;
    hence contradiction by A7,A6,Th37,XREAL_1:157;
  end;
  then reconsider r =s-t as Element of NAT by INT_1:5;
  1=m*s-m*t by A4,A5;
  then 1=m*r;
  hence contradiction by A1,NAT_1:15;
end;
