let S be non empty non void ManySortedSign ; :: thesis: for U0 being MSAlgebra over S
for U1, U2 being MSSubAlgebra of U0 st the Sorts of U1 c= the Sorts of U2 holds
U1 is MSSubAlgebra of U2

let U0 be MSAlgebra over S; :: thesis: for U1, U2 being MSSubAlgebra of U0 st the Sorts of U1 c= the Sorts of U2 holds
U1 is MSSubAlgebra of U2

let U1, U2 be MSSubAlgebra of U0; :: thesis: ( the Sorts of U1 c= the Sorts of U2 implies U1 is MSSubAlgebra of U2 )
reconsider B1 = the Sorts of U1, B2 = the Sorts of U2 as MSSubset of U0 by Def9;
assume A1: the Sorts of U1 c= the Sorts of U2 ; :: thesis: U1 is MSSubAlgebra of U2
hence the Sorts of U1 is MSSubset of U2 by PBOOLE:def 18; :: according to MSUALG_2:def 9 :: thesis: for B being MSSubset of U2 st B = the Sorts of U1 holds
( B is opers_closed & the Charact of U1 = Opers (U2,B) )

let B be MSSubset of U2; :: thesis: ( B = the Sorts of U1 implies ( B is opers_closed & the Charact of U1 = Opers (U2,B) ) )
A2: B1 is opers_closed by Def9;
set O = the Charact of U1;
set P = Opers (U2,B);
A3: the Charact of U1 = Opers (U0,B1) by Def9;
A4: B2 is opers_closed by Def9;
A5: the Charact of U2 = Opers (U0,B2) by Def9;
assume A6: B = the Sorts of U1 ; :: thesis: ( B is opers_closed & the Charact of U1 = Opers (U2,B) )
A7: for o being OperSymbol of S holds B is_closed_on o
proof
let o be OperSymbol of S; :: thesis:
A8: B1 is_closed_on o by A2;
A9: B2 is_closed_on o by A4;
A10: Den (o,U2) = (Opers (U0,B2)) . o by
.= o /. B2 by Def8
.= (Den (o,U0)) | (((B2 #) * the Arity of S) . o) by ;
Den (o,U1) = (Opers (U0,B1)) . o by
.= o /. B1 by Def8
.= (Den (o,U0)) | (((B1 #) * the Arity of S) . o) by
.= (Den (o,U0)) | ((((B2 #) * the Arity of S) . o) /\ (((B1 #) * the Arity of S) . o)) by
.= (Den (o,U2)) | (((B1 #) * the Arity of S) . o) by ;
then rng ((Den (o,U2)) | (((B1 #) * the Arity of S) . o)) c= Result (o,U1) by RELAT_1:def 19;
then rng ((Den (o,U2)) | (((B1 #) * the Arity of S) . o)) c= ( the Sorts of U1 * the ResultSort of S) . o by MSUALG_1:def 5;
hence B is_closed_on o by A6; :: thesis: verum
end;
hence B is opers_closed ; :: thesis: the Charact of U1 = Opers (U2,B)
for x being object st x in the carrier' of S holds
the Charact of U1 . x = (Opers (U2,B)) . x
proof
let x be object ; :: thesis: ( x in the carrier' of S implies the Charact of U1 . x = (Opers (U2,B)) . x )
assume x in the carrier' of S ; :: thesis: the Charact of U1 . x = (Opers (U2,B)) . x
then reconsider o = x as OperSymbol of S ;
A11: B1 is_closed_on o by A2;
A12: B2 is_closed_on o by A4;
A13: Den (o,U2) = (Opers (U0,B2)) . o by
.= o /. B2 by Def8
.= (Den (o,U0)) | (((B2 #) * the Arity of S) . o) by ;
thus the Charact of U1 . x = o /. B1 by
.= (Den (o,U0)) | (((B1 #) * the Arity of S) . o) by
.= (Den (o,U0)) | ((((B2 #) * the Arity of S) . o) /\ (((B1 #) * the Arity of S) . o)) by
.= (Den (o,U2)) | (((B1 #) * the Arity of S) . o) by
.= o /. B by A6, A7, Def7
.= (Opers (U2,B)) . x by Def8 ; :: thesis: verum
end;
hence the Charact of U1 = Opers (U2,B) ; :: thesis: verum