reserve V for non empty RLSStruct;
reserve x,y,y1 for set;
reserve v for VECTOR of V;
reserve a,b for Real;
reserve V for non empty addLoopStr;
reserve F for FinSequence-like PartFunc of NAT,V;
reserve f,f9,g for sequence of V;
reserve v,u for Element of V;
reserve j,k,n for Nat;
reserve V for RealLinearSpace;
reserve v for VECTOR of V;
reserve F,G,H,I for FinSequence of V;
reserve V for add-associative right_zeroed right_complementable non empty
  addLoopStr;
reserve F for FinSequence of V;
reserve v,v1,v2,u,w for Element of V;
reserve j,k for Nat;

theorem Th60:
  for V being RealLinearSpace, v being VECTOR of V holds Sum<* v,v *> = 2 * v
proof
  let V be RealLinearSpace, v be VECTOR of V;
  thus Sum<* v,v *> = v + v by Th45
    .= 1 * v + v by Def8
    .= 1 * v + 1 * v by Def8
    .= (1 + 1) * v by Def6
    .= 2 * v;
end;
