
theorem ThSL18:
  for L being Z_Lattice, L1 being Z_Sublattice of L,
  w1, w2 being Vector of L1, v1, v2 being Vector of L
  st w1 = v1 & w2 = v2 holds <; w1, w2 ;> = <; v1, v2 ;>
  proof
    let L be Z_Lattice, L1 be Z_Sublattice of L,
    w1, w2 be Vector of L1, v1, v2 be Vector of L such that
    B1: w1 = v1 & w2 = v2;
    B2: [w1, w2] in [:the carrier of L1, the carrier of L1:];
    thus <; w1, w2 ;> = ((the scalar of L) || the carrier of L1).(w1, w2)
    by defSublattice
    .= <; v1, v2 ;> by B1,B2,FUNCT_1:49;
  end;
