reserve a,b,c for Integer;
reserve i,j,k,l for Nat;
reserve n for Nat;
reserve a,b,c,d,a1,b1,a2,b2,k,l for Integer;

theorem Th25:
  c divides a*b & a,c are_coprime implies c divides b
proof
  assume that
A1: c divides a*b and
A2: a,c are_coprime;
  c divides c*b;
  then
A3: c divides (a*b gcd c*b) by A1,Def2;
A4: a*b gcd c*b = |.b.| by A2,Th24;
  b<0 implies c divides b
  proof
    assume b<0;
    then c divides (-b) by A4,A3,ABSVALUE:def 1;
    hence thesis by Th10;
  end;
  hence thesis by A4,A3,ABSVALUE:def 1;
end;
