reserve n for Nat,
  a,b,c,d for Real,
  s for Real_Sequence;

theorem
  (a+b+c+d)|^2 = a|^2+b|^2+c|^2+d|^2+(2*a*b+2*a*c+2*a*d) +(2*b*c+2*b*d)+ 2*c*d
proof
  (a+b+c+d)|^(1+1) = (a+b+c+d)|^1*(a+b+c+d) by NEWTON:6
    .=(a+b+c+d)*(a+b+c+d)
    .=(a*a+a*b+a*c+a*d)+(a*b+b*b+b*c+b*d)+ (a*c+c*b+c*c+c*d)+(a*d+d*b+d*c+d*
  d)
    .=(a*a+a*b+a*c+a*d)+(a*b+b|^2+b*c+b*d)+ (a*c+c*b+c*c+c*d)+(a*d+d*b+d*c+d
  *d) by WSIERP_1:1
    .=(a|^2+a*b+a*c+a*d)+(a*b+b|^2+b*c+b*d)+ (a*c+c*b+c*c+c*d)+(a*d+d*b+d*c+
  d*d) by WSIERP_1:1
    .=(a|^2+a*b+a*c+a*d)+(a*b+b|^2+b*c+b*d)+ (a*c+c*b+c|^2+c*d)+(a*d+d*b+d*c
  +d*d) by WSIERP_1:1
    .=(a|^2+a*b+a*c+a*d)+(a*b+b|^2+b*c+b*d)+ (a*c+c*b+c|^2+c*d)+(a*d+d*b+d*c
  +d|^2) by WSIERP_1:1
    .=a|^2+c|^2+b|^2+d|^2+(2*a*b+2*a*c+2*a*d)+(2*b*c+2*b*d)+2*c*d;
  hence thesis;
end;
