reserve x,y for object;
reserve S for non void non empty ManySortedSign,
  o for OperSymbol of S,
  U0,U1, U2 for MSAlgebra over S;
reserve U0 for non-empty MSAlgebra over S;

theorem Th31:
  for S be non void non empty ManySortedSign, U0 be non-empty
  MSAlgebra over S holds MSAlg_meet(U0) is commutative
proof
  let S be non void non empty ManySortedSign, U0 be non-empty MSAlgebra over S;
  set o = MSAlg_meet(U0);
  for x,y be Element of MSSub(U0) holds o.(x,y)=o.(y,x)
  proof
    let x,y be Element of MSSub(U0);
    reconsider U1=x,U2=y as strict MSSubAlgebra of U0 by Def19;
A1: the Sorts of(U2 /\ U1) = (the Sorts of U2) (/\) (the Sorts of U1) &
   for B2 be MSSubset of U0 st B2=the Sorts of (U2/\U1)
   holds B2 is opers_closed & the Charact of (U2/\U1) = Opers(U0,B2) by Def16;
    o.(x,y) = U1 /\ U2 & o.(y,x) = U2 /\ U1 by Def21;
    hence thesis by A1,Def16;
  end;
  hence thesis by BINOP_1:def 2;
end;
