reserve L for Ortholattice,
  a, b, c for Element of L;

theorem
  a _|_ b & a _|_ c implies a _|_ (b "/\" c) & a _|_ (b "\/" c)
proof
  assume a _|_ b;
  then
A1: a [= b`;
  then
A2: a "/\" c` [= b` "/\" c` by LATTICES:9;
  assume
A3: a _|_ c;
  b`[= b` "\/" c` by LATTICES:5;
  then a [= b` "\/" c` by A1,LATTICES:7;
  then a [= (b` "\/" c`)`` by ROBBINS3:def 6;
  then a [= (b "/\" c)` by ROBBINS1:def 23;
  hence a _|_ (b "/\" c);
  a [= c` by A3;
  then a [= b` "/\" c` by A2,LATTICES:4;
  then a [= (b`` "\/" c``)` by ROBBINS1:def 23;
  then a [= (b "\/" c``)` by ROBBINS3:def 6;
  then a [= (b "\/" c)` by ROBBINS3:def 6;
  hence thesis;
end;
