let L be with_infima Poset; :: thesis: for X being Subset of L st ( ex_inf_of X,L or L is complete ) holds
inf X = inf (fininfs X)
let X be Subset of L; :: thesis: ( ( ex_inf_of X,L or L is complete ) implies inf X = inf (fininfs X) )
assume ( ex_inf_of X,L or L is complete ) ; :: thesis: inf X = inf (fininfs X)
then A1: ex_inf_of X,L by YELLOW_0:17;

A2: now :: thesis: for Y being finite Subset of X st Y <> {} holds
ex_inf_of Y,L

let Y be finite Subset of X; :: thesis: ( Y <> {} implies ex_inf_of Y,L )
Y c= the carrier of L by XBOOLE_1:1;
hence ( Y <> {} implies ex_inf_of Y,L ) by YELLOW_0:55; :: thesis: verum

end;

A3: now :: thesis: for x being Element of L st x in fininfs X holds
ex Y being finite Subset of X st
( ex_inf_of Y,L & x = "/\" (Y,L) )

let x be Element of L; :: thesis: ( x in fininfs X implies ex Y being finite Subset of X st
( ex_inf_of Y,L & x = "/\" (Y,L) ) )
assume x in fininfs X ; :: thesis: ex Y being finite Subset of X st
( ex_inf_of Y,L & x = "/\" (Y,L) )
then ex Y being finite Subset of X st
( x = "/\" (Y,L) & ex_inf_of Y,L ) ;
hence ex Y being finite Subset of X st
( ex_inf_of Y,L & x = "/\" (Y,L) ) ; :: thesis: verum

end;

now :: thesis: for Y being finite Subset of X st Y <> {} holds
"/\" (Y,L) in fininfs X

let Y be finite Subset of X; :: thesis: ( Y <> {} implies "/\" (Y,L) in fininfs X )
reconsider Z = Y as Subset of L by XBOOLE_1:1;
assume Y <> {} ; :: thesis: "/\" (Y,L) in fininfs X
then ex_inf_of Z,L by YELLOW_0:55;
hence "/\" (Y,L) in fininfs X ; :: thesis: verum

end;

hence inf X = inf (fininfs X) by A1, A2, A3, Th59; :: thesis: verum