reserve i,j,k,n for Nat;
reserve x,y,z for Tuple of n, BOOLEAN;

theorem Th5:
  for a,b,c being Nat holds 0<a & 0<b & a<c & b<c & c is
  prime implies (a * b) mod c <> 0
proof
  let a,b,c be Nat;
  assume that
A1: 0<a & 0<b and
A2: a<c and
A3: b<c and
A4: c is prime;
  assume (a * b) mod c = 0;
  then (a * b) = c * ((a * b) div c) + 0 by A2,NAT_D:2;
  then c divides (a * b) by NAT_D:def 3;
  then c divides a or c divides b by A4,NEWTON:80;
  hence contradiction by A1,A2,A3,NAT_D:7;
end;
