reserve i for Nat,
  j for Element of NAT,
  X,Y,x,y,z for set;
reserve C for initialized ConstructorSignature,
  s for SortSymbol of C,
  o for OperSymbol of C,
  c for constructor OperSymbol of C;
reserve a,b for expression of C, an_Adj C;
reserve t, t1,t2 for expression of C, a_Type C;
reserve p for FinSequence of QuasiTerms C;
reserve e for expression of C;
reserve a,a9 for expression of C, an_Adj C;
reserve q for pure expression of C, a_Type C,
  A for finite Subset of QuasiAdjs C;
reserve T for quasi-type of C;

theorem Th111:
  for V1, V2 being Element of VarPoset holds
  V1 "\/" V2 = V1 /\ V2 & V1 "/\" V2 = V1 \/ V2
proof
  let V1, V2 be Element of VarPoset;
  set V = the set of all varcl A where A is finite Subset of Vars;
  set A0 = the finite Subset of Vars;
  varcl A0 in V;
  then reconsider V as non empty set;
A1: VarPoset = (InclPoset V) opp;
A2: the carrier of InclPoset V = V by YELLOW_1:1;
  reconsider v1 = V1, v2 = V2 as Element of (InclPoset V) opp;
  reconsider a1 = V1, a2 = V2 as Element of InclPoset V;
  V1 in V by A2;
  then consider A1 being finite Subset of Vars such that
A3: V1 = varcl A1;
  V2 in V by A2;
  then consider A2 being finite Subset of Vars such that
A4: V2 = varcl A2;
A5: a1~ = v1;
A6: a2~ = v2;
A7: InclPoset V is with_infima with_suprema by A1,LATTICE3:10,YELLOW_7:16;
  reconsider x1 = V1, x2 = V2 as finite Subset of Vars by A3,A4,Th24;
  V1 /\ V2 = varcl (x1 /\ x2) by A3,A4,Th13;
  then V1 /\ V2 in V;
  then a1 "/\" a2 = V1 /\ V2 by YELLOW_1:9;
  hence V1 "\/" V2 = V1 /\ V2 by A5,A6,A7,YELLOW_7:21;
  V1 \/ V2 = varcl (A1 \/ A2) by A3,A4,Th11;
  then a1 \/ a2 in V;
  then a1 "\/" a2 = V1 \/ V2 by YELLOW_1:8;
  hence thesis by A5,A6,A7,YELLOW_7:23;
end;
