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

theorem
  (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;
