
theorem Th1:
  for L being lower-bounded sup-Semilattice
  for AR being auxiliary(ii) auxiliary(iii) Relation of L
  for x, y, z, u being Element of L holds
  [x, z] in AR & [y, u] in AR implies [x "\/" y, z "\/" u] in AR
proof
  let L be lower-bounded sup-Semilattice;
  let AR be auxiliary(ii) auxiliary(iii) Relation of L;
  let x,y,z,u be Element of L;
  assume that
A1: [x, z] in AR and
A2: [y, u] in AR;
A3: x <= x;
A4: y <= y;
A5: z <= z "\/" u by YELLOW_0:22;
A6: u <= z "\/" u by YELLOW_0:22;
A7: [x, z "\/" u] in AR by A1,A3,A5,Def4;
  [y, z "\/" u] in AR by A2,A4,A6,Def4;
  hence thesis by A7,Def5;
end;
