let L be complete Lattice; :: thesis: for f being monotone UnOp of L holds

( lfp f is_a_fixpoint_of f & ex O being Ordinal st

( card O c= card the carrier of L & (f,O) +. (Bottom L) = lfp f ) )

let f be monotone UnOp of L; :: thesis: ( lfp f is_a_fixpoint_of f & ex O being Ordinal st

( card O c= card the carrier of L & (f,O) +. (Bottom L) = lfp f ) )

reconsider ze = Bottom L as Element of L ;

A1: Bottom L [= f . ze by LATTICES:16;

then consider O being Ordinal such that

A2: card O c= card the carrier of L and

A3: (f,O) +. (Bottom L) is_a_fixpoint_of f by Th30;

card the carrier of L in nextcard the carrier of L by CARD_1:def 3;

then card O in nextcard the carrier of L by A2, ORDINAL1:12;

then O in nextcard the carrier of L by CARD_3:44;

then A4: O c= nextcard the carrier of L by ORDINAL1:def 2;

hence lfp f is_a_fixpoint_of f by A1, A3, Th28; :: thesis: ex O being Ordinal st

( card O c= card the carrier of L & (f,O) +. (Bottom L) = lfp f )

take O ; :: thesis: ( card O c= card the carrier of L & (f,O) +. (Bottom L) = lfp f )

thus card O c= card the carrier of L by A2; :: thesis: (f,O) +. (Bottom L) = lfp f

thus (f,O) +. (Bottom L) = lfp f by A1, A3, A4, Th28; :: thesis: verum

( lfp f is_a_fixpoint_of f & ex O being Ordinal st

( card O c= card the carrier of L & (f,O) +. (Bottom L) = lfp f ) )

let f be monotone UnOp of L; :: thesis: ( lfp f is_a_fixpoint_of f & ex O being Ordinal st

( card O c= card the carrier of L & (f,O) +. (Bottom L) = lfp f ) )

reconsider ze = Bottom L as Element of L ;

A1: Bottom L [= f . ze by LATTICES:16;

then consider O being Ordinal such that

A2: card O c= card the carrier of L and

A3: (f,O) +. (Bottom L) is_a_fixpoint_of f by Th30;

card the carrier of L in nextcard the carrier of L by CARD_1:def 3;

then card O in nextcard the carrier of L by A2, ORDINAL1:12;

then O in nextcard the carrier of L by CARD_3:44;

then A4: O c= nextcard the carrier of L by ORDINAL1:def 2;

hence lfp f is_a_fixpoint_of f by A1, A3, Th28; :: thesis: ex O being Ordinal st

( card O c= card the carrier of L & (f,O) +. (Bottom L) = lfp f )

take O ; :: thesis: ( card O c= card the carrier of L & (f,O) +. (Bottom L) = lfp f )

thus card O c= card the carrier of L by A2; :: thesis: (f,O) +. (Bottom L) = lfp f

thus (f,O) +. (Bottom L) = lfp f by A1, A3, A4, Th28; :: thesis: verum