
theorem Th30:
  for p,q being 2_greater Prime
  st p <> q
  holds Leg(p,q) * Leg(q,p) = (-1)|^( ((p-1)/2) * ((q-1)/2) )
proof
let p,q be 2_greater Prime;
assume A1:  p <> q;
A2: p > 2 & q > 2 by Def1;
p - 1 > 2 - 1 by Def1,XREAL_1:9;
then p -' 1 = p - 1 by NAT_D:39;
then A3: (p-'1) div 2 = (p-1)/2;
q - 1 > 2 - 1 by Def1,XREAL_1:9;
then q -' 1 = q - 1 by NAT_D:39; then
A4: (-1)|^(((p-'1) div 2)*((q-'1) div 2))
  = (-1)|^( ((p-1)/2) * ((q-1)/2) ) by A3;
thus Leg(p,q) * Leg(q,p) = Leg(p,q) * Lege(q,p) by Lm4
      .= Lege(p,q) * Lege(q,p) by Lm4
      .= (-1)|^( ((p-1)/2) * ((q-1)/2) ) by A1,A2,A4,INT_5:49;
end;
