
theorem Th58:
  for K be add-associative right_zeroed right_complementable non
  empty addLoopStr for V be non empty ModuleStr over K for f be alternating
additiveSAF additiveFAF Form of V,V for v,w be Vector of V holds f.(v,w)=-f.(w,
  v)
proof
  let K be add-associative right_zeroed right_complementable non empty
  addLoopStr, V be non empty ModuleStr over K, f be alternating additiveSAF
  additiveFAF Form of V,V, v,w be Vector of V;
  0.K = f.(v+w,v+w) by Def26
    .= f.(v,v) + f.(v,w) + (f.(w,v) + f.(w,w)) by Th28
    .= 0.K + f.(v,w) + (f.(w,v) + f.(w,w)) by Def26
    .= 0.K + f.(v,w) + (f.(w,v) + 0.K) by Def26
    .= 0.K + f.(v,w) + f.(w,v) by RLVECT_1:def 4
    .= f.(v,w) + f.(w,v) by RLVECT_1:4;
  hence thesis by RLVECT_1:6;
end;
