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

theorem Th1:
  (a+b+c)|^2=a|^2+b|^2+c|^2+2*a*b+2*a*c+2*b*c
proof
  (a+b+c)|^(1+1) = (a+b+c)|^1*(a+b+c) by NEWTON:6
    .= (a+b+c)*(a+b+c)
    .= (a*a+a*b+a*c)+(a*b+b*b+b*c)+(a*c+c*b+c*c)
    .= (a*a+a*b+a*c)+(a*b+b|^2+b*c)+(a*c+c*b+c*c) by WSIERP_1:1
    .= (a|^2+a*b+a*c)+(a*b+b|^2+b*c)+(a*c+c*b+c*c) by WSIERP_1:1
    .= (a|^2+a*b+a*c)+(a*b+b|^2+b*c)+(a*c+c*b+c|^2) by WSIERP_1:1
    .= a|^2+c|^2+2*a*b+2*a*c+2*b*c+b|^2;
  hence thesis;
end;
