let U0 be with_const_op Universal_Algebra; :: thesis: for U1, U2 being strict SubAlgebra of U0 holds U1 /\ (U1 "\/" U2) = U1
let U1, U2 be strict SubAlgebra of U0; :: thesis: U1 /\ (U1 "\/" U2) = U1
reconsider u112 = the carrier of (U1 /\ (U1 "\/" U2)) as non empty Subset of U0 by Def7;
reconsider u1 = the carrier of U1, u2 = the carrier of U2 as non empty Subset of U0 by Def7;
reconsider A = u1 \/ u2 as non empty Subset of U0 ;
A1: the charact of U1 = Opers (U0,u1) by Def7;
A2: dom (Opers (U0,u1)) = dom the charact of U0 by Def6;
U1 "\/" U2 = GenUnivAlg A by Def13;
then A3: A c= the carrier of (U1 "\/" U2) by Def12;
A4: the carrier of U1 meets the carrier of (U1 "\/" U2) by Th17;
then A5: the carrier of (U1 /\ (U1 "\/" U2)) = the carrier of U1 /\ the carrier of (U1 "\/" U2) by Def9;
then A6: the carrier of (U1 /\ (U1 "\/" U2)) c= the carrier of U1 by XBOOLE_1:17;
the carrier of U1 c= A by XBOOLE_1:7;
then the carrier of U1 c= the carrier of (U1 "\/" U2) by A3;
then A7: the carrier of U1 c= the carrier of (U1 /\ (U1 "\/" U2)) by ;
A8: dom (Opers (U0,u112)) = dom the charact of U0 by Def6;
A9: for n being Nat st n in dom the charact of U0 holds
the charact of (U1 /\ (U1 "\/" U2)) . n = the charact of U1 . n
proof
let n be Nat; :: thesis: ( n in dom the charact of U0 implies the charact of (U1 /\ (U1 "\/" U2)) . n = the charact of U1 . n )
assume A10: n in dom the charact of U0 ; :: thesis: the charact of (U1 /\ (U1 "\/" U2)) . n = the charact of U1 . n
then reconsider o0 = the charact of U0 . n as operation of U0 by FUNCT_1:def 3;
thus the charact of (U1 /\ (U1 "\/" U2)) . n = (Opers (U0,u112)) . n by
.= o0 /. u112 by
.= o0 /. u1 by
.= (Opers (U0,u1)) . n by
.= the charact of U1 . n by Def7 ; :: thesis: verum
end;
the charact of (U1 /\ (U1 "\/" U2)) = Opers (U0,u112) by ;
then the charact of (U1 /\ (U1 "\/" U2)) = the charact of U1 by ;
hence U1 /\ (U1 "\/" U2) = U1 by ; :: thesis: verum