reserve G for Robbins join-associative join-commutative non empty
  ComplLLattStr;
reserve x, y, z, u, v for Element of G;

theorem Th48: :: Lemma 3
  \delta (x _3, x) = x _0
proof
  set alpha = -x _3 + x _1 + -Double x;
  x = Expand (-x _3 + x _0, x) by Th36
    .= \delta (-x _3 + x _1, -(\delta (x _3, x _0) + x)) by LATTICES:def 5
    .= \delta (-x _3 + x _1, -Double x) by Th45;
  then
A1: -Double x = \delta (-x _3 + x _1 + -Double x, x) by Th36;
A2: x = \delta (Double x, x _0) by Th40
    .= \delta (-alpha + x, x _0) by A1;
  -x _3 = Expand (x _1 + -Double x, -x _3) by Th36
    .= \delta (-x _3 + x _1 + -Double x, \delta (x _1 + -Double x, -x _3))
  by LATTICES:def 5
    .= \delta (alpha, \delta (x _0 + (x + Double x), \delta (Double x, x _1)
  )) by Th47
    .= \delta (alpha, \delta (Double x + x _1, \delta (Double x, x _1))) by
LATTICES:def 5
    .= -(-alpha + x _1) by Th36;
  hence \delta (x _3, x) = \delta (-alpha + (x _0 + x), x)
    .= Expand (-alpha + x, x _0) by A2,LATTICES:def 5
    .= x _0 by Th36;
end;
