reserve x,X for set,
        r,r1,r2,s for Real,
        i,j,k,m,n for Nat;

theorem Th3:
  for f be real-valued FinSequence
    for i st i in dom f holds Sum sqr(f+*(i,r)) = (Sum sqr f)-(f.i)^2+r^2
proof
  let f be real-valued FinSequence;
  let i such that
  A1: i in dom f;
  reconsider fi=f.i as Element of REAL by XREAL_0:def 1;
  set F=@@f;
  set G=F| (i-' 1),H=F/^i;
  A2: sqr<*fi*>=<*fi^2*> by RVSUM_1:55;
  F=F+*(i,fi) by FUNCT_7:35
  .=G^<*fi*>^H by A1,FUNCT_7:98;
  then sqr F=sqr(G^<*fi*>)^sqr H by RVSUM_1:144
           .=sqr G^sqr<*fi*>^sqr H by RVSUM_1:144;
  then A3: Sum sqr F=Sum(sqr G^sqr<*fi*>)+Sum sqr H by RVSUM_1:75
   .=Sum sqr G+fi^2+Sum sqr H by A2,RVSUM_1:74;
  reconsider R=r as Element of REAL by XREAL_0:def 1;
  A4: sqr<*R*>=<*R^2*> by RVSUM_1:55;
  F+*(i,R)=G^<*R*>^H by A1,FUNCT_7:98;
  then sqr(F+*(i,R))=sqr(G^<*R*>)^sqr H by RVSUM_1:144
   .=sqr G^sqr<*R*>^sqr H by RVSUM_1:144;
  then Sum sqr(F+*(i,R))=Sum(sqr G^sqr<*R*>)+Sum sqr H by RVSUM_1:75
   .=Sum sqr G+R^2+Sum sqr H by A4,RVSUM_1:74;
  hence thesis by A3;
end;
