reserve X for set;
reserve a,b,c,k,m,n for Nat;
reserve i,j for Integer;
reserve r,s for Real;
reserve p,p1,p2,p3 for Prime;

theorem Th102:
  for x,y being Complex st
   x = (3|^n-3 to_power(1-n)-2)/4 & y = (3|^n+3 to_power(1-n)-4)/8
   holds x*(x+1) = 4*y*(y+1)
  proof
    let x,y be Complex;
    set a = 3 to_power(1-n);
    set b = 3|^n;
    assume that
A1: x = (b-a-2)/4 and
A2: y = (b+a-4)/8;
    b = 3 to_power n;
    then a*b = 3 to_power (1-n+n) by POWER:27;
    then
A3: 4*(a*b-3) = 0;
    thus x*(x+1) = (b-a-2)*(b-a+2)/(4*4) by A1
    .= (b+a-4)*((b+a+4)/(2*8)) by A3
    .= 4*y*(y+1) by A2;
  end;
