let S, T be complete Scott TopLattice; :: thesis: UPS S,T = SCMaps S,T
A1: the carrier of (UPS S,T) = the carrier of (SCMaps S,T)

proof

thus the carrier of (UPS S,T) c= the carrier of (SCMaps S,T) :: according to XBOOLE_0:def 10 :: thesis: the carrier of (SCMaps S,T) c= the carrier of (UPS S,T)

proof

let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in the carrier of (UPS S,T) or x in the carrier of (SCMaps S,T) )
assume x in the carrier of (UPS S,T) ; :: thesis: x in the carrier of (SCMaps S,T)
then reconsider f = x as directed-sups-preserving Function of S,T by Def4;
f is continuous ;
hence x in the carrier of (SCMaps S,T) by WAYBEL17:def 2; :: thesis: verum

end;

let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in the carrier of (SCMaps S,T) or x in the carrier of (UPS S,T) )
assume A2: x in the carrier of (SCMaps S,T) ; :: thesis: x in the carrier of (UPS S,T)
the carrier of (SCMaps S,T) c= the carrier of (MonMaps S,T) by YELLOW_0:def 13;
then reconsider f = x as Function of S,T by A2, WAYBEL10:10;
f is continuous by A2, WAYBEL17:def 2;
hence x in the carrier of (UPS S,T) by Def4; :: thesis: verum

end;

then A3: the carrier of (UPS S,T) c= the carrier of (MonMaps S,T) by YELLOW_0:def 13;
UPS S,T is full SubRelStr of T |^ the carrier of S by Def4;
then the InternalRel of (UPS S,T) = the InternalRel of (T |^ the carrier of S) |_2 the carrier of (UPS S,T) by YELLOW_0:def 14
.= (the InternalRel of (T |^ the carrier of S) |_2 the carrier of (MonMaps S,T)) |_2 the carrier of (UPS S,T) by A3, WELLORD1:29
.= the InternalRel of (MonMaps S,T) |_2 the carrier of (SCMaps S,T) by A1, YELLOW_0:def 14
.= the InternalRel of (SCMaps S,T) by YELLOW_0:def 14 ;
hence UPS S,T = SCMaps S,T by A1; :: thesis: verum