reserve x for set,
  R for non empty Poset;
reserve S1 for OrderSortedSign,
  OU0 for OSAlgebra of S1;
reserve s,s1,s2,s3,s4 for SortSymbol of S1;
reserve U0 for non-empty OSAlgebra of S1;

theorem Th47:
  OSAlg_meet(U0) is associative
proof
  set o = OSAlg_meet(U0);
  set m = MSAlg_meet(U0);
A1: m is associative by MSUALG_2:32;
  for x,y,z be Element of OSSub(U0) holds o.(x,o.(y,z))=o.(o.(x,y),z)
  proof
    let x,y,z be Element of OSSub(U0);
A2: o.(x,y) = m.(x,y) by Th43;
    o.(y,z) = m.(y,z) by Th43;
    then o.(x,o.(y,z)) = m.(x,m.(y,z)) by Th43
      .= m.(m.(x,y),z) by A1,BINOP_1:def 3
      .= o.(o.(x,y),z) by A2,Th43;
    hence thesis;
  end;
  hence thesis by BINOP_1:def 3;
end;
