reserve n,i,j,k,l for Nat;
reserve D for non empty set;
reserve c,d for Element of D;
reserve p,q,q9,r for FinSequence of D;
reserve RAS for MidSp-like non empty ReperAlgebraStr over n+2;
reserve a,b,d,pii,p9i for Point of RAS;
reserve p,q for Tuple of (n+1),RAS;
reserve m for Nat of n;
reserve W for ATLAS of RAS;
reserve v for Vector of W;
reserve x,y for Tuple of (n+1),W;

theorem Th13:
  (for l being Nat of n st l=i holds (x+*(i,v)).l = v) & for l,i
  being Nat of n st l<>i holds (x+*(i,v)).l = x.l
proof
  thus for l being Nat of n st l=i holds (x+*(i,v)).l = v
  proof
    let l be Nat of n such that
A1: l = i;
    l in Seg(n+1) by Th7;
    hence thesis by A1,Lm1;
  end;
  thus thesis by FUNCT_7:32;
end;
