let Y be set ; :: thesis: for r being Real
for h being PartFunc of REAL,REAL holds
( ( h | Y is increasing & 0 < r implies (r (#) h) | Y is increasing ) & ( r = 0 implies (r (#) h) | Y is V8() ) & ( h | Y is increasing & r < 0 implies (r (#) h) | Y is decreasing ) )

let r be Real; :: thesis: for h being PartFunc of REAL,REAL holds
( ( h | Y is increasing & 0 < r implies (r (#) h) | Y is increasing ) & ( r = 0 implies (r (#) h) | Y is V8() ) & ( h | Y is increasing & r < 0 implies (r (#) h) | Y is decreasing ) )

let h be PartFunc of REAL,REAL; :: thesis: ( ( h | Y is increasing & 0 < r implies (r (#) h) | Y is increasing ) & ( r = 0 implies (r (#) h) | Y is V8() ) & ( h | Y is increasing & r < 0 implies (r (#) h) | Y is decreasing ) )
thus ( h | Y is increasing & 0 < r implies (r (#) h) | Y is increasing ) :: thesis: ( ( r = 0 implies (r (#) h) | Y is V8() ) & ( h | Y is increasing & r < 0 implies (r (#) h) | Y is decreasing ) )
proof
assume that
A1: h | Y is increasing and
A2: 0 < r ; :: thesis: (r (#) h) | Y is increasing
now :: thesis: for r1, r2 being Real st r1 in Y /\ (dom (r (#) h)) & r2 in Y /\ (dom (r (#) h)) & r1 < r2 holds
(r (#) h) . r1 < (r (#) h) . r2
let r1, r2 be Real; :: thesis: ( r1 in Y /\ (dom (r (#) h)) & r2 in Y /\ (dom (r (#) h)) & r1 < r2 implies (r (#) h) . r1 < (r (#) h) . r2 )
assume that
A3: r1 in Y /\ (dom (r (#) h)) and
A4: r2 in Y /\ (dom (r (#) h)) and
A5: r1 < r2 ; :: thesis: (r (#) h) . r1 < (r (#) h) . r2
A6: r2 in Y by ;
A7: r2 in dom (r (#) h) by ;
then r2 in dom h by VALUED_1:def 5;
then A8: r2 in Y /\ (dom h) by ;
A9: r1 in Y by ;
A10: r1 in dom (r (#) h) by ;
then r1 in dom h by VALUED_1:def 5;
then r1 in Y /\ (dom h) by ;
then h . r1 < h . r2 by A1, A5, A8, Th20;
then r * (h . r1) < r * (h . r2) by ;
then (r (#) h) . r1 < r * (h . r2) by ;
hence (r (#) h) . r1 < (r (#) h) . r2 by ; :: thesis: verum
end;
hence (r (#) h) | Y is increasing by Th20; :: thesis: verum
end;
thus ( r = 0 implies (r (#) h) | Y is V8() ) :: thesis: ( h | Y is increasing & r < 0 implies (r (#) h) | Y is decreasing )
proof
assume A11: r = 0 ; :: thesis: (r (#) h) | Y is V8()
A12: 0 in REAL by XREAL_0:def 1;
now :: thesis: for r1 being Element of REAL st r1 in Y /\ (dom (r (#) h)) holds
(r (#) h) . r1 = 0
let r1 be Element of REAL ; :: thesis: ( r1 in Y /\ (dom (r (#) h)) implies (r (#) h) . r1 = 0 )
assume r1 in Y /\ (dom (r (#) h)) ; :: thesis: (r (#) h) . r1 = 0
then A13: r1 in dom (r (#) h) by XBOOLE_0:def 4;
r * (h . r1) = 0 by A11;
hence (r (#) h) . r1 = 0 by ; :: thesis: verum
end;
hence (r (#) h) | Y is V8() by ; :: thesis: verum
end;
assume that
A14: h | Y is increasing and
A15: r < 0 ; :: thesis: (r (#) h) | Y is decreasing
now :: thesis: for r1, r2 being Real st r1 in Y /\ (dom (r (#) h)) & r2 in Y /\ (dom (r (#) h)) & r1 < r2 holds
(r (#) h) . r2 < (r (#) h) . r1
let r1, r2 be Real; :: thesis: ( r1 in Y /\ (dom (r (#) h)) & r2 in Y /\ (dom (r (#) h)) & r1 < r2 implies (r (#) h) . r2 < (r (#) h) . r1 )
assume that
A16: r1 in Y /\ (dom (r (#) h)) and
A17: r2 in Y /\ (dom (r (#) h)) and
A18: r1 < r2 ; :: thesis: (r (#) h) . r2 < (r (#) h) . r1
A19: r2 in Y by ;
A20: r2 in dom (r (#) h) by ;
then r2 in dom h by VALUED_1:def 5;
then A21: r2 in Y /\ (dom h) by ;
A22: r1 in Y by ;
A23: r1 in dom (r (#) h) by ;
then r1 in dom h by VALUED_1:def 5;
then r1 in Y /\ (dom h) by ;
then h . r1 < h . r2 by ;
then r * (h . r2) < r * (h . r1) by ;
then (r (#) h) . r2 < r * (h . r1) by ;
hence (r (#) h) . r2 < (r (#) h) . r1 by ; :: thesis: verum
end;
hence (r (#) h) | Y is decreasing by Th21; :: thesis: verum