reserve a,b,c,k,l,m,n for Nat,
  i,j,x,y for Integer;

theorem Th6:
  a <> 0 & c <> 0 & a*c gcd b*c = c implies a,b are_coprime
proof
  assume that
A1: a <> 0 and
A2: c <> 0 and
A3: a*c gcd b*c = c;
  a*c <> 0*c by A1,A2,XCMPLX_1:5;
  then consider a1,b1 being Integer such that
A4: a*c = (a*c gcd b*c)*a1 and
A5: b*c = (a*c gcd b*c)*b1 & a1,b1 are_coprime by INT_2:23;
  a = a1 by A2,A3,A4,XCMPLX_1:5;
  hence thesis by A2,A3,A5,XCMPLX_1:5;
end;
