Article MATHMORP, MML version 4.99.1005
:: MATHMORP:funcnot 1 => MATHMORP:func 1
definition
let a1 be Element of NAT;
let a2 be Element of the carrier of TOP-REAL a1;
let a3 be Element of bool the carrier of TOP-REAL a1;
func A3 + A2 -> Element of bool the carrier of TOP-REAL a1 equals
{b1 + a2 where b1 is Element of the carrier of TOP-REAL a1: b1 in a3};
end;
:: MATHMORP:def 1
theorem
for b1 being Element of NAT
for b2 being Element of the carrier of TOP-REAL b1
for b3 being Element of bool the carrier of TOP-REAL b1 holds
b3 + b2 = {b4 + b2 where b4 is Element of the carrier of TOP-REAL b1: b4 in b3};
:: MATHMORP:funcnot 2 => MATHMORP:func 2
definition
let a1 be Element of NAT;
let a2 be Element of bool the carrier of TOP-REAL a1;
func A2 ! -> Element of bool the carrier of TOP-REAL a1 equals
{- b1 where b1 is Element of the carrier of TOP-REAL a1: b1 in a2};
end;
:: MATHMORP:def 2
theorem
for b1 being Element of NAT
for b2 being Element of bool the carrier of TOP-REAL b1 holds
b2 ! = {- b3 where b3 is Element of the carrier of TOP-REAL b1: b3 in b2};
:: MATHMORP:funcnot 3 => MATHMORP:func 3
definition
let a1 be Element of NAT;
let a2, a3 be Element of bool the carrier of TOP-REAL a1;
func A2 (-) A3 -> Element of bool the carrier of TOP-REAL a1 equals
{b1 where b1 is Element of the carrier of TOP-REAL a1: a3 + b1 c= a2};
end;
:: MATHMORP:def 3
theorem
for b1 being Element of NAT
for b2, b3 being Element of bool the carrier of TOP-REAL b1 holds
b2 (-) b3 = {b4 where b4 is Element of the carrier of TOP-REAL b1: b3 + b4 c= b2};
:: MATHMORP:funcnot 4 => MATHMORP:func 4
definition
let a1 be Element of NAT;
let a2, a3 be Element of bool the carrier of TOP-REAL a1;
func A2 (+) A3 -> Element of bool the carrier of TOP-REAL a1 equals
{b1 + b2 where b1 is Element of the carrier of TOP-REAL a1, b2 is Element of the carrier of TOP-REAL a1: b1 in a2 & b2 in a3};
end;
:: MATHMORP:def 4
theorem
for b1 being Element of NAT
for b2, b3 being Element of bool the carrier of TOP-REAL b1 holds
b2 (+) b3 = {b4 + b5 where b4 is Element of the carrier of TOP-REAL b1, b5 is Element of the carrier of TOP-REAL b1: b4 in b2 & b5 in b3};
:: MATHMORP:th 1
theorem
for b1 being Element of NAT
for b2 being Element of bool the carrier of TOP-REAL b1 holds
b2 ! ! = b2;
:: MATHMORP:th 2
theorem
for b1 being Element of NAT
for b2 being Element of the carrier of TOP-REAL b1 holds
{0.REAL b1} + b2 = {b2};
:: MATHMORP:th 3
theorem
for b1 being Element of NAT
for b2, b3 being Element of bool the carrier of TOP-REAL b1
for b4 being Element of the carrier of TOP-REAL b1
st b2 c= b3
holds b2 + b4 c= b3 + b4;
:: MATHMORP:th 4
theorem
for b1 being Element of NAT
for b2 being Element of the carrier of TOP-REAL b1
for b3 being Element of bool the carrier of TOP-REAL b1
st b3 = {}
holds b3 + b2 = {};
:: MATHMORP:th 5
theorem
for b1 being Element of NAT
for b2 being Element of bool the carrier of TOP-REAL b1 holds
b2 (-) {0.REAL b1} = b2;
:: MATHMORP:th 6
theorem
for b1 being Element of NAT
for b2 being Element of bool the carrier of TOP-REAL b1 holds
b2 (+) {0.REAL b1} = b2;
:: MATHMORP:th 7
theorem
for b1 being Element of NAT
for b2 being Element of bool the carrier of TOP-REAL b1
for b3 being Element of the carrier of TOP-REAL b1 holds
b2 (+) {b3} = b2 + b3;
:: MATHMORP:th 8
theorem
for b1 being Element of NAT
for b2, b3 being Element of bool the carrier of TOP-REAL b1
st b3 = {}
holds b2 (-) b3 = REAL b1;
:: MATHMORP:th 9
theorem
for b1 being Element of NAT
for b2, b3, b4 being Element of bool the carrier of TOP-REAL b1
st b2 c= b3
holds b2 (-) b4 c= b3 (-) b4 & b2 (+) b4 c= b3 (+) b4;
:: MATHMORP:th 10
theorem
for b1 being Element of NAT
for b2, b3, b4 being Element of bool the carrier of TOP-REAL b1
st b2 c= b3
holds b4 (-) b3 c= b4 (-) b2 & b4 (+) b2 c= b4 (+) b3;
:: MATHMORP:th 11
theorem
for b1 being Element of NAT
for b2, b3 being Element of bool the carrier of TOP-REAL b1
st 0.REAL b1 in b2
holds b3 (-) b2 c= b3 & b3 c= b3 (+) b2;
:: MATHMORP:th 12
theorem
for b1 being Element of NAT
for b2, b3 being Element of bool the carrier of TOP-REAL b1 holds
b2 (+) b3 = b3 (+) b2;
:: MATHMORP:th 13
theorem
for b1 being Element of NAT
for b2, b3 being Element of bool the carrier of TOP-REAL b1
for b4, b5 being Element of the carrier of TOP-REAL b1 holds
b2 + b4 c= b3 + b5
iff
b2 + (b4 - b5) c= b3;
:: MATHMORP:th 14
theorem
for b1 being Element of NAT
for b2, b3 being Element of bool the carrier of TOP-REAL b1
for b4 being Element of the carrier of TOP-REAL b1 holds
(b2 + b4) (-) b3 = (b2 (-) b3) + b4;
:: MATHMORP:th 15
theorem
for b1 being Element of NAT
for b2, b3 being Element of bool the carrier of TOP-REAL b1
for b4 being Element of the carrier of TOP-REAL b1 holds
(b2 + b4) (+) b3 = (b2 (+) b3) + b4;
:: MATHMORP:th 16
theorem
for b1 being Element of NAT
for b2 being Element of bool the carrier of TOP-REAL b1
for b3, b4 being Element of the carrier of TOP-REAL b1 holds
(b2 + b3) + b4 = b2 + (b3 + b4);
:: MATHMORP:th 17
theorem
for b1 being Element of NAT
for b2, b3 being Element of bool the carrier of TOP-REAL b1
for b4 being Element of the carrier of TOP-REAL b1 holds
b2 (-) (b3 + b4) = (b2 (-) b3) + - b4;
:: MATHMORP:th 18
theorem
for b1 being Element of NAT
for b2, b3 being Element of bool the carrier of TOP-REAL b1
for b4 being Element of the carrier of TOP-REAL b1 holds
b2 (+) (b3 + b4) = (b2 (+) b3) + b4;
:: MATHMORP:th 19
theorem
for b1 being Element of NAT
for b2, b3 being Element of bool the carrier of TOP-REAL b1
for b4 being Element of the carrier of TOP-REAL b1
st b4 in b2
holds b3 + b4 c= b3 (+) b2;
:: MATHMORP:th 20
theorem
for b1 being Element of NAT
for b2, b3 being Element of bool the carrier of TOP-REAL b1 holds
b2 c= (b2 (+) b3) (-) b3;
:: MATHMORP:th 21
theorem
for b1 being Element of NAT
for b2 being Element of bool the carrier of TOP-REAL b1 holds
b2 + 0.REAL b1 = b2;
:: MATHMORP:th 22
theorem
for b1 being Element of NAT
for b2 being Element of bool the carrier of TOP-REAL b1
for b3 being Element of the carrier of TOP-REAL b1 holds
b2 (-) {b3} = b2 + - b3;
:: MATHMORP:th 23
theorem
for b1 being Element of NAT
for b2, b3, b4 being Element of bool the carrier of TOP-REAL b1 holds
b2 (-) (b3 (+) b4) = (b2 (-) b3) (-) b4;
:: MATHMORP:th 24
theorem
for b1 being Element of NAT
for b2, b3, b4 being Element of bool the carrier of TOP-REAL b1 holds
b2 (-) (b3 (+) b4) = (b2 (-) b4) (-) b3;
:: MATHMORP:th 25
theorem
for b1 being Element of NAT
for b2, b3, b4 being Element of bool the carrier of TOP-REAL b1 holds
b2 (+) (b3 (-) b4) c= (b2 (+) b3) (-) b4;
:: MATHMORP:th 26
theorem
for b1 being Element of NAT
for b2, b3, b4 being Element of bool the carrier of TOP-REAL b1 holds
b2 (+) (b3 (+) b4) = (b2 (+) b3) (+) b4;
:: MATHMORP:th 27
theorem
for b1 being Element of NAT
for b2, b3 being Element of bool the carrier of TOP-REAL b1
for b4 being Element of the carrier of TOP-REAL b1 holds
(b2 \/ b3) + b4 = (b2 + b4) \/ (b3 + b4);
:: MATHMORP:th 28
theorem
for b1 being Element of NAT
for b2, b3 being Element of bool the carrier of TOP-REAL b1
for b4 being Element of the carrier of TOP-REAL b1 holds
(b2 /\ b3) + b4 = (b2 + b4) /\ (b3 + b4);
:: MATHMORP:th 29
theorem
for b1 being Element of NAT
for b2, b3, b4 being Element of bool the carrier of TOP-REAL b1 holds
b2 (-) (b3 \/ b4) = (b2 (-) b3) /\ (b2 (-) b4);
:: MATHMORP:th 30
theorem
for b1 being Element of NAT
for b2, b3, b4 being Element of bool the carrier of TOP-REAL b1 holds
b2 (+) (b3 \/ b4) = (b2 (+) b3) \/ (b2 (+) b4);
:: MATHMORP:th 31
theorem
for b1 being Element of NAT
for b2, b3, b4 being Element of bool the carrier of TOP-REAL b1 holds
(b2 (-) b3) \/ (b4 (-) b3) c= (b2 \/ b4) (-) b3;
:: MATHMORP:th 32
theorem
for b1 being Element of NAT
for b2, b3, b4 being Element of bool the carrier of TOP-REAL b1 holds
(b2 \/ b3) (+) b4 = (b2 (+) b4) \/ (b3 (+) b4);
:: MATHMORP:th 33
theorem
for b1 being Element of NAT
for b2, b3, b4 being Element of bool the carrier of TOP-REAL b1 holds
(b2 /\ b3) (-) b4 = (b2 (-) b4) /\ (b3 (-) b4);
:: MATHMORP:th 34
theorem
for b1 being Element of NAT
for b2, b3, b4 being Element of bool the carrier of TOP-REAL b1 holds
(b2 /\ b3) (+) b4 c= (b2 (+) b4) /\ (b3 (+) b4);
:: MATHMORP:th 35
theorem
for b1 being Element of NAT
for b2, b3, b4 being Element of bool the carrier of TOP-REAL b1 holds
b2 (+) (b3 /\ b4) c= (b2 (+) b3) /\ (b2 (+) b4);
:: MATHMORP:th 36
theorem
for b1 being Element of NAT
for b2, b3, b4 being Element of bool the carrier of TOP-REAL b1 holds
(b2 (-) b3) \/ (b2 (-) b4) c= b2 (-) (b3 /\ b4);
:: MATHMORP:th 37
theorem
for b1 being Element of NAT
for b2, b3 being Element of bool the carrier of TOP-REAL b1 holds
(b2 ` (-) b3) ` = b2 (+) (b3 !);
:: MATHMORP:th 38
theorem
for b1 being Element of NAT
for b2, b3 being Element of bool the carrier of TOP-REAL b1 holds
(b2 (-) b3) ` = b2 ` (+) (b3 !);
:: MATHMORP:funcnot 5 => MATHMORP:func 5
definition
let a1 be Element of NAT;
let a2, a3 be Element of bool the carrier of TOP-REAL a1;
func A2 (O) A3 -> Element of bool the carrier of TOP-REAL a1 equals
(a2 (-) a3) (+) a3;
end;
:: MATHMORP:def 5
theorem
for b1 being Element of NAT
for b2, b3 being Element of bool the carrier of TOP-REAL b1 holds
b2 (O) b3 = (b2 (-) b3) (+) b3;
:: MATHMORP:funcnot 6 => MATHMORP:func 6
definition
let a1 be Element of NAT;
let a2, a3 be Element of bool the carrier of TOP-REAL a1;
func A2 (o) A3 -> Element of bool the carrier of TOP-REAL a1 equals
(a2 (+) a3) (-) a3;
end;
:: MATHMORP:def 6
theorem
for b1 being Element of NAT
for b2, b3 being Element of bool the carrier of TOP-REAL b1 holds
b2 (o) b3 = (b2 (+) b3) (-) b3;
:: MATHMORP:th 39
theorem
for b1 being Element of NAT
for b2, b3 being Element of bool the carrier of TOP-REAL b1 holds
(b2 ` (O) (b3 !)) ` = b2 (o) b3;
:: MATHMORP:th 40
theorem
for b1 being Element of NAT
for b2, b3 being Element of bool the carrier of TOP-REAL b1 holds
(b2 ` (o) (b3 !)) ` = b2 (O) b3;
:: MATHMORP:th 41
theorem
for b1 being Element of NAT
for b2, b3 being Element of bool the carrier of TOP-REAL b1 holds
b2 (O) b3 c= b2 & b2 c= b2 (o) b3;
:: MATHMORP:th 42
theorem
for b1 being Element of NAT
for b2 being Element of bool the carrier of TOP-REAL b1 holds
b2 (O) b2 = b2;
:: MATHMORP:th 43
theorem
for b1 being Element of NAT
for b2, b3 being Element of bool the carrier of TOP-REAL b1 holds
(b2 (O) b3) (-) b3 c= b2 (-) b3 & (b2 (O) b3) (+) b3 c= b2 (+) b3;
:: MATHMORP:th 44
theorem
for b1 being Element of NAT
for b2, b3 being Element of bool the carrier of TOP-REAL b1 holds
b2 (-) b3 c= (b2 (o) b3) (-) b3 & b2 (+) b3 c= (b2 (o) b3) (+) b3;
:: MATHMORP:th 45
theorem
for b1 being Element of NAT
for b2, b3, b4 being Element of bool the carrier of TOP-REAL b1
st b2 c= b3
holds b2 (O) b4 c= b3 (O) b4 & b2 (o) b4 c= b3 (o) b4;
:: MATHMORP:th 46
theorem
for b1 being Element of NAT
for b2, b3 being Element of bool the carrier of TOP-REAL b1
for b4 being Element of the carrier of TOP-REAL b1 holds
(b2 + b4) (O) b3 = (b2 (O) b3) + b4;
:: MATHMORP:th 47
theorem
for b1 being Element of NAT
for b2, b3 being Element of bool the carrier of TOP-REAL b1
for b4 being Element of the carrier of TOP-REAL b1 holds
(b2 + b4) (o) b3 = (b2 (o) b3) + b4;
:: MATHMORP:th 48
theorem
for b1 being Element of NAT
for b2, b3, b4 being Element of bool the carrier of TOP-REAL b1
st b2 c= b3
holds b4 (O) b3 c= (b4 (-) b2) (+) b3;
:: MATHMORP:th 49
theorem
for b1 being Element of NAT
for b2, b3, b4 being Element of bool the carrier of TOP-REAL b1
st b2 c= b3
holds b4 (o) b2 c= (b4 (+) b3) (-) b2;
:: MATHMORP:th 50
theorem
for b1 being Element of NAT
for b2, b3 being Element of bool the carrier of TOP-REAL b1 holds
b2 (+) b3 = (b2 (o) b3) (+) b3 & b2 (-) b3 = (b2 (O) b3) (-) b3;
:: MATHMORP:th 51
theorem
for b1 being Element of NAT
for b2, b3 being Element of bool the carrier of TOP-REAL b1 holds
b2 (+) b3 = (b2 (+) b3) (O) b3 & b2 (-) b3 = (b2 (-) b3) (o) b3;
:: MATHMORP:th 52
theorem
for b1 being Element of NAT
for b2, b3 being Element of bool the carrier of TOP-REAL b1 holds
(b2 (O) b3) (O) b3 = b2 (O) b3;
:: MATHMORP:th 53
theorem
for b1 being Element of NAT
for b2, b3 being Element of bool the carrier of TOP-REAL b1 holds
(b2 (o) b3) (o) b3 = b2 (o) b3;
:: MATHMORP:th 54
theorem
for b1 being Element of NAT
for b2, b3, b4 being Element of bool the carrier of TOP-REAL b1 holds
b2 (O) b3 c= (b2 \/ b4) (O) b3;
:: MATHMORP:th 55
theorem
for b1 being Element of NAT
for b2, b3, b4 being Element of bool the carrier of TOP-REAL b1
st b2 = b2 (O) b3
holds b4 (O) b2 c= b4 (O) b3;
:: MATHMORP:funcnot 7 => MATHMORP:func 7
definition
let a1 be real set;
let a2 be Element of NAT;
let a3 be Element of bool the carrier of TOP-REAL a2;
func A1 (.) A3 -> Element of bool the carrier of TOP-REAL a2 equals
{a1 * b1 where b1 is Element of the carrier of TOP-REAL a2: b1 in a3};
end;
:: MATHMORP:def 7
theorem
for b1 being real set
for b2 being Element of NAT
for b3 being Element of bool the carrier of TOP-REAL b2 holds
b1 (.) b3 = {b1 * b4 where b4 is Element of the carrier of TOP-REAL b2: b4 in b3};
:: MATHMORP:th 56
theorem
for b1 being Element of NAT
for b2 being Element of bool the carrier of TOP-REAL b1
st b2 = {}
holds 0 (.) b2 = {};
:: MATHMORP:th 57
theorem
for b1 being Element of NAT
for b2 being non empty Element of bool the carrier of TOP-REAL b1 holds
0 (.) b2 = {0.REAL b1};
:: MATHMORP:th 58
theorem
for b1 being Element of NAT
for b2 being Element of bool the carrier of TOP-REAL b1 holds
1 (.) b2 = b2;
:: MATHMORP:th 59
theorem
for b1 being Element of NAT
for b2 being Element of bool the carrier of TOP-REAL b1 holds
2 (.) b2 c= b2 (+) b2;
:: MATHMORP:th 60
theorem
for b1 being Element of NAT
for b2 being Element of bool the carrier of TOP-REAL b1
for b3, b4 being real set holds
(b3 * b4) (.) b2 = b3 (.) (b4 (.) b2);
:: MATHMORP:th 61
theorem
for b1 being Element of NAT
for b2, b3 being Element of bool the carrier of TOP-REAL b1
for b4 being real set
st b2 c= b3
holds b4 (.) b2 c= b4 (.) b3;
:: MATHMORP:th 62
theorem
for b1 being Element of NAT
for b2 being Element of bool the carrier of TOP-REAL b1
for b3 being Element of the carrier of TOP-REAL b1
for b4 being real set holds
b4 (.) (b2 + b3) = (b4 (.) b2) + (b4 * b3);
:: MATHMORP:th 63
theorem
for b1 being Element of NAT
for b2, b3 being Element of bool the carrier of TOP-REAL b1
for b4 being real set holds
b4 (.) (b2 (+) b3) = (b4 (.) b2) (+) (b4 (.) b3);
:: MATHMORP:th 64
theorem
for b1 being Element of NAT
for b2, b3 being Element of bool the carrier of TOP-REAL b1
for b4 being real set
st b4 <> 0
holds b4 (.) (b2 (-) b3) = (b4 (.) b2) (-) (b4 (.) b3);
:: MATHMORP:th 65
theorem
for b1 being Element of NAT
for b2, b3 being Element of bool the carrier of TOP-REAL b1
for b4 being real set
st b4 <> 0
holds b4 (.) (b2 (O) b3) = (b4 (.) b2) (O) (b4 (.) b3);
:: MATHMORP:th 66
theorem
for b1 being Element of NAT
for b2, b3 being Element of bool the carrier of TOP-REAL b1
for b4 being real set
st b4 <> 0
holds b4 (.) (b2 (o) b3) = (b4 (.) b2) (o) (b4 (.) b3);
:: MATHMORP:funcnot 8 => MATHMORP:func 8
definition
let a1 be Element of NAT;
let a2, a3, a4 be Element of bool the carrier of TOP-REAL a1;
func A2 (*)(A3,A4) -> Element of bool the carrier of TOP-REAL a1 equals
(a2 (-) a3) /\ (a2 ` (-) a4);
end;
:: MATHMORP:def 8
theorem
for b1 being Element of NAT
for b2, b3, b4 being Element of bool the carrier of TOP-REAL b1 holds
b2 (*)(b3,b4) = (b2 (-) b3) /\ (b2 ` (-) b4);
:: MATHMORP:funcnot 9 => MATHMORP:func 9
definition
let a1 be Element of NAT;
let a2, a3, a4 be Element of bool the carrier of TOP-REAL a1;
func A2 (&)(A3,A4) -> Element of bool the carrier of TOP-REAL a1 equals
a2 \/ (a2 (*)(a3,a4));
end;
:: MATHMORP:def 9
theorem
for b1 being Element of NAT
for b2, b3, b4 being Element of bool the carrier of TOP-REAL b1 holds
b2 (&)(b3,b4) = b2 \/ (b2 (*)(b3,b4));
:: MATHMORP:funcnot 10 => MATHMORP:func 10
definition
let a1 be Element of NAT;
let a2, a3, a4 be Element of bool the carrier of TOP-REAL a1;
func A2 (@)(A3,A4) -> Element of bool the carrier of TOP-REAL a1 equals
a2 \ (a2 (*)(a3,a4));
end;
:: MATHMORP:def 10
theorem
for b1 being Element of NAT
for b2, b3, b4 being Element of bool the carrier of TOP-REAL b1 holds
b2 (@)(b3,b4) = b2 \ (b2 (*)(b3,b4));
:: MATHMORP:th 67
theorem
for b1 being Element of NAT
for b2, b3, b4 being Element of bool the carrier of TOP-REAL b1
st b2 = {}
holds b3 (*)(b2,b4) = b3 ` (-) b4;
:: MATHMORP:th 68
theorem
for b1 being Element of NAT
for b2, b3, b4 being Element of bool the carrier of TOP-REAL b1
st b2 = {}
holds b3 (*)(b4,b2) = b3 (-) b4;
:: MATHMORP:th 69
theorem
for b1 being Element of NAT
for b2, b3, b4 being Element of bool the carrier of TOP-REAL b1
st 0.REAL b1 in b2
holds b3 (*)(b2,b4) c= b3;
:: MATHMORP:th 70
theorem
for b1 being Element of NAT
for b2, b3, b4 being Element of bool the carrier of TOP-REAL b1
st 0.REAL b1 in b2
holds (b3 (*)(b4,b2)) /\ b3 = {};
:: MATHMORP:th 71
theorem
for b1 being Element of NAT
for b2, b3, b4 being Element of bool the carrier of TOP-REAL b1
st 0.REAL b1 in b2
holds b3 (&)(b2,b4) = b3;
:: MATHMORP:th 72
theorem
for b1 being Element of NAT
for b2, b3, b4 being Element of bool the carrier of TOP-REAL b1
st 0.REAL b1 in b2
holds b3 (@)(b4,b2) = b3;
:: MATHMORP:th 74
theorem
for b1 being Element of NAT
for b2, b3, b4 being Element of bool the carrier of TOP-REAL b1 holds
b2 (@)(b3,b4) = (b2 ` (&)(b4,b3)) `;
:: MATHMORP:th 75
theorem
for b1 being Element of NAT
for b2 being Element of bool the carrier of TOP-REAL b1 holds
b2 is convex(b1)
iff
for b3, b4 being Element of the carrier of TOP-REAL b1
for b5 being real set
st 0 <= b5 & b5 <= 1 & b3 in b2 & b4 in b2
holds (b5 * b3) + ((1 - b5) * b4) in b2;
:: MATHMORP:attrnot 1 => JORDAN1:attr 1
definition
let a1 be Element of NAT;
let a2 be Element of bool the carrier of TOP-REAL a1;
attr a2 is convex means
for b1, b2 being Element of the carrier of TOP-REAL a1
for b3 being real set
st 0 <= b3 & b3 <= 1 & b1 in a2 & b2 in a2
holds (b3 * b1) + ((1 - b3) * b2) in a2;
end;
:: MATHMORP:dfs 11
definiens
let a1 be Element of NAT;
let a2 be Element of bool the carrier of TOP-REAL a1;
To prove
a2 is convex
it is sufficient to prove
thus for b1, b2 being Element of the carrier of TOP-REAL a1
for b3 being real set
st 0 <= b3 & b3 <= 1 & b1 in a2 & b2 in a2
holds (b3 * b1) + ((1 - b3) * b2) in a2;
:: MATHMORP:def 11
theorem
for b1 being Element of NAT
for b2 being Element of bool the carrier of TOP-REAL b1 holds
b2 is convex(b1)
iff
for b3, b4 being Element of the carrier of TOP-REAL b1
for b5 being real set
st 0 <= b5 & b5 <= 1 & b3 in b2 & b4 in b2
holds (b5 * b3) + ((1 - b5) * b4) in b2;
:: MATHMORP:th 76
theorem
for b1 being Element of NAT
for b2 being Element of bool the carrier of TOP-REAL b1
st b2 is convex(b1)
holds b2 ! is convex(b1);
:: MATHMORP:th 77
theorem
for b1 being Element of NAT
for b2, b3 being Element of bool the carrier of TOP-REAL b1
st b2 is convex(b1) & b3 is convex(b1)
holds b2 (+) b3 is convex(b1) & b2 (-) b3 is convex(b1);
:: MATHMORP:th 78
theorem
for b1 being Element of NAT
for b2, b3 being Element of bool the carrier of TOP-REAL b1
st b2 is convex(b1) & b3 is convex(b1)
holds b2 (O) b3 is convex(b1) & b2 (o) b3 is convex(b1);
:: MATHMORP:th 79
theorem
for b1 being Element of NAT
for b2 being Element of bool the carrier of TOP-REAL b1
for b3, b4 being real set
st b2 is convex(b1) & 0 < b3 & 0 < b4
holds (b4 + b3) (.) b2 = (b4 (.) b2) (+) (b3 (.) b2);