reserve a,b,c,k,m,n for Nat;
reserve i,j,x,y for Integer;
reserve p,q for Prime;
reserve r,s for Real;

theorem Th67:
  for m,n being positive Nat, s being natural Number
  st Polygon(s,m) = Polygon(s,n) & s >= 2 holds m = n
  proof
    let m,n be positive Nat;
    let s be natural Number;
    assume Polygon(s,m) = Polygon(s,n);
    then m^2*(s-2) - m*(s-4) - n^2*(s-2) + n*(s-4) = 0;
    then (m-n) * ((s-2)*(m+n-1)+2) = 0;
    then
A1: m-n = 0 or (s-2)*(m+n-1)+2 = 0;
    assume
A2: s >= 2;
    now
      assume
A3:   (s-2)*(m+n-1)+2 = 0;
      s-2 >= 2-2 by A2,XREAL_1:6;
      hence contradiction by A3;
    end;
    hence thesis by A1;
  end;
