let Y be set ; :: thesis: for r being Real

for h being PartFunc of REAL,REAL holds

( ( h | Y is non-decreasing & 0 <= r implies (r (#) h) | Y is non-decreasing ) & ( h | Y is non-decreasing & r <= 0 implies (r (#) h) | Y is non-increasing ) )

let r be Real; :: thesis: for h being PartFunc of REAL,REAL holds

( ( h | Y is non-decreasing & 0 <= r implies (r (#) h) | Y is non-decreasing ) & ( h | Y is non-decreasing & r <= 0 implies (r (#) h) | Y is non-increasing ) )

let h be PartFunc of REAL,REAL; :: thesis: ( ( h | Y is non-decreasing & 0 <= r implies (r (#) h) | Y is non-decreasing ) & ( h | Y is non-decreasing & r <= 0 implies (r (#) h) | Y is non-increasing ) )

thus ( h | Y is non-decreasing & 0 <= r implies (r (#) h) | Y is non-decreasing ) :: thesis: ( h | Y is non-decreasing & r <= 0 implies (r (#) h) | Y is non-increasing )

A11: h | Y is non-decreasing and

A12: r <= 0 ; :: thesis: (r (#) h) | Y is non-increasing

for h being PartFunc of REAL,REAL holds

( ( h | Y is non-decreasing & 0 <= r implies (r (#) h) | Y is non-decreasing ) & ( h | Y is non-decreasing & r <= 0 implies (r (#) h) | Y is non-increasing ) )

let r be Real; :: thesis: for h being PartFunc of REAL,REAL holds

( ( h | Y is non-decreasing & 0 <= r implies (r (#) h) | Y is non-decreasing ) & ( h | Y is non-decreasing & r <= 0 implies (r (#) h) | Y is non-increasing ) )

let h be PartFunc of REAL,REAL; :: thesis: ( ( h | Y is non-decreasing & 0 <= r implies (r (#) h) | Y is non-decreasing ) & ( h | Y is non-decreasing & r <= 0 implies (r (#) h) | Y is non-increasing ) )

thus ( h | Y is non-decreasing & 0 <= r implies (r (#) h) | Y is non-decreasing ) :: thesis: ( h | Y is non-decreasing & r <= 0 implies (r (#) h) | Y is non-increasing )

proof

assume that
assume that

A1: h | Y is non-decreasing and

A2: 0 <= r ; :: thesis: (r (#) h) | Y is non-decreasing

end;A1: h | Y is non-decreasing and

A2: 0 <= r ; :: thesis: (r (#) h) | Y is non-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) . r1 <= (r (#) h) . r2

hence
(r (#) h) | Y is non-decreasing
by Th22; :: thesis: verum(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 A4, XBOOLE_0:def 4;

A7: r2 in dom (r (#) h) by A4, XBOOLE_0:def 4;

then r2 in dom h by VALUED_1:def 5;

then A8: r2 in Y /\ (dom h) by A6, XBOOLE_0:def 4;

A9: r1 in Y by A3, XBOOLE_0:def 4;

A10: r1 in dom (r (#) h) by A3, XBOOLE_0:def 4;

then r1 in dom h by VALUED_1:def 5;

then r1 in Y /\ (dom h) by A9, XBOOLE_0:def 4;

then h . r1 <= h . r2 by A1, A5, A8, Th22;

then r * (h . r1) <= (h . r2) * r by A2, XREAL_1:64;

then (r (#) h) . r1 <= r * (h . r2) by A10, VALUED_1:def 5;

hence (r (#) h) . r1 <= (r (#) h) . r2 by A7, VALUED_1:def 5; :: thesis: verum

end;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 A4, XBOOLE_0:def 4;

A7: r2 in dom (r (#) h) by A4, XBOOLE_0:def 4;

then r2 in dom h by VALUED_1:def 5;

then A8: r2 in Y /\ (dom h) by A6, XBOOLE_0:def 4;

A9: r1 in Y by A3, XBOOLE_0:def 4;

A10: r1 in dom (r (#) h) by A3, XBOOLE_0:def 4;

then r1 in dom h by VALUED_1:def 5;

then r1 in Y /\ (dom h) by A9, XBOOLE_0:def 4;

then h . r1 <= h . r2 by A1, A5, A8, Th22;

then r * (h . r1) <= (h . r2) * r by A2, XREAL_1:64;

then (r (#) h) . r1 <= r * (h . r2) by A10, VALUED_1:def 5;

hence (r (#) h) . r1 <= (r (#) h) . r2 by A7, VALUED_1:def 5; :: thesis: verum

A11: h | Y is non-decreasing and

A12: r <= 0 ; :: thesis: (r (#) h) | Y is non-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) . r2 <= (r (#) h) . r1

hence
(r (#) h) | Y is non-increasing
by Th23; :: thesis: verum(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

A13: r1 in Y /\ (dom (r (#) h)) and

A14: r2 in Y /\ (dom (r (#) h)) and

A15: r1 < r2 ; :: thesis: (r (#) h) . r2 <= (r (#) h) . r1

A16: r2 in Y by A14, XBOOLE_0:def 4;

A17: r2 in dom (r (#) h) by A14, XBOOLE_0:def 4;

then r2 in dom h by VALUED_1:def 5;

then A18: r2 in Y /\ (dom h) by A16, XBOOLE_0:def 4;

A19: r1 in Y by A13, XBOOLE_0:def 4;

A20: r1 in dom (r (#) h) by A13, XBOOLE_0:def 4;

then r1 in dom h by VALUED_1:def 5;

then r1 in Y /\ (dom h) by A19, XBOOLE_0:def 4;

then h . r1 <= h . r2 by A11, A15, A18, Th22;

then r * (h . r2) <= r * (h . r1) by A12, XREAL_1:65;

then (r (#) h) . r2 <= r * (h . r1) by A17, VALUED_1:def 5;

hence (r (#) h) . r2 <= (r (#) h) . r1 by A20, VALUED_1:def 5; :: thesis: verum

end;assume that

A13: r1 in Y /\ (dom (r (#) h)) and

A14: r2 in Y /\ (dom (r (#) h)) and

A15: r1 < r2 ; :: thesis: (r (#) h) . r2 <= (r (#) h) . r1

A16: r2 in Y by A14, XBOOLE_0:def 4;

A17: r2 in dom (r (#) h) by A14, XBOOLE_0:def 4;

then r2 in dom h by VALUED_1:def 5;

then A18: r2 in Y /\ (dom h) by A16, XBOOLE_0:def 4;

A19: r1 in Y by A13, XBOOLE_0:def 4;

A20: r1 in dom (r (#) h) by A13, XBOOLE_0:def 4;

then r1 in dom h by VALUED_1:def 5;

then r1 in Y /\ (dom h) by A19, XBOOLE_0:def 4;

then h . r1 <= h . r2 by A11, A15, A18, Th22;

then r * (h . r2) <= r * (h . r1) by A12, XREAL_1:65;

then (r (#) h) . r2 <= r * (h . r1) by A17, VALUED_1:def 5;

hence (r (#) h) . r2 <= (r (#) h) . r1 by A20, VALUED_1:def 5; :: thesis: verum