let V1, V2 be Element of VarPoset; :: thesis: ( V1 "\/" V2 = V1 /\ V2 & V1 "/\" V2 = V1 \/ V2 )
set V = { (varcl A) where A is finite Subset of Vars : verum } ;
set A0 = the finite Subset of Vars;
varcl the finite Subset of Vars in { (varcl A) where A is finite Subset of Vars : verum } ;
then reconsider V = { (varcl A) where A is finite Subset of Vars : verum } 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 & InclPoset V is 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; :: thesis: V1 "/\" V2 = V1 \/ V2
V1 \/ V2 = varcl (A1 \/ A2) by A3, A4, Th11;
then a1 \/ a2 in V ;
then a1 "\/" a2 = V1 \/ V2 by YELLOW_1:8;
hence V1 "/\" V2 = V1 \/ V2 by A5, A6, A7, YELLOW_7:23; :: thesis: verum