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
  for a,c be Nat, b be non zero Nat holds
  a*b < c < a*(b+1) implies not a divides c & not c divides a
  proof
    let a,c be Nat, b be non zero Nat;
    assume
    A1: a*b < c < a*(b+1); then
    reconsider c as non zero Nat;
    reconsider a as non zero Nat by A1;
    assume not thesis; then
    per cases;
    suppose a divides c;
      then consider k be Nat such that
      B1: c = a*k by NAT_D:def 3;
      b < k by A1,B1,XREAL_1:64; then
      b + 1 <= k by NAT_1:13;
      hence contradiction by A1,B1,XREAL_1:64;
    end;
    suppose
      B1: c divides a;
      a*b >= a*1 by XREAL_1:64,NAT_1:14; then
      c > a by A1,XXREAL_0:2;
      hence thesis by B1,NAT_D:7;
    end;
  end;
