Article EUCLID_3, MML version 4.99.1005
:: EUCLID_3:funcnot 1 => EUCLID_3:func 1
definition
let a1 be Element of COMPLEX;
func cpx2euc A1 -> Element of the carrier of TOP-REAL 2 equals
|[Re a1,Im a1]|;
end;
:: EUCLID_3:def 1
theorem
for b1 being Element of COMPLEX holds
cpx2euc b1 = |[Re b1,Im b1]|;
:: EUCLID_3:funcnot 2 => EUCLID_3:func 2
definition
let a1 be Element of the carrier of TOP-REAL 2;
func euc2cpx A1 -> Element of COMPLEX equals
[*a1 `1,a1 `2*];
end;
:: EUCLID_3:def 2
theorem
for b1 being Element of the carrier of TOP-REAL 2 holds
euc2cpx b1 = [*b1 `1,b1 `2*];
:: EUCLID_3:th 1
theorem
for b1 being Element of COMPLEX holds
euc2cpx cpx2euc b1 = b1;
:: EUCLID_3:th 2
theorem
for b1 being Element of the carrier of TOP-REAL 2 holds
cpx2euc euc2cpx b1 = b1;
:: EUCLID_3:th 3
theorem
for b1 being Element of the carrier of TOP-REAL 2 holds
ex b2 being Element of COMPLEX st
b1 = cpx2euc b2;
:: EUCLID_3:th 4
theorem
for b1 being Element of COMPLEX holds
ex b2 being Element of the carrier of TOP-REAL 2 st
b1 = euc2cpx b2;
:: EUCLID_3:th 5
theorem
for b1, b2 being Element of COMPLEX
st cpx2euc b1 = cpx2euc b2
holds b1 = b2;
:: EUCLID_3:th 6
theorem
for b1, b2 being Element of the carrier of TOP-REAL 2
st euc2cpx b1 = euc2cpx b2
holds b1 = b2;
:: EUCLID_3:th 8
theorem
for b1 being Element of the carrier of TOP-REAL 2 holds
Re euc2cpx b1 = b1 `1 & Im euc2cpx b1 = b1 `2;
:: EUCLID_3:th 9
theorem
for b1, b2 being Element of REAL holds
cpx2euc [*b1,b2*] = |[b1,b2]|;
:: EUCLID_3:th 10
theorem
for b1, b2 being Element of COMPLEX holds
|[Re (b1 + b2),Im (b1 + b2)]| = |[(Re b1) + Re b2,(Im b1) + Im b2]|;
:: EUCLID_3:th 11
theorem
for b1, b2 being Element of COMPLEX holds
cpx2euc (b1 + b2) = (cpx2euc b1) + cpx2euc b2;
:: EUCLID_3:th 12
theorem
for b1, b2 being Element of the carrier of TOP-REAL 2 holds
[*(b1 + b2) `1,(b1 + b2) `2*] = [*b1 `1 + (b2 `1),b1 `2 + (b2 `2)*];
:: EUCLID_3:th 13
theorem
for b1, b2 being Element of the carrier of TOP-REAL 2 holds
euc2cpx (b1 + b2) = (euc2cpx b1) + euc2cpx b2;
:: EUCLID_3:th 14
theorem
for b1 being Element of COMPLEX holds
|[Re - b1,Im - b1]| = |[- Re b1,- Im b1]|;
:: EUCLID_3:th 15
theorem
for b1 being Element of COMPLEX holds
cpx2euc - b1 = - cpx2euc b1;
:: EUCLID_3:th 16
theorem
for b1 being Element of the carrier of TOP-REAL 2 holds
[*(- b1) `1,(- b1) `2*] = [*- (b1 `1),- (b1 `2)*];
:: EUCLID_3:th 17
theorem
for b1 being Element of the carrier of TOP-REAL 2 holds
euc2cpx - b1 = - euc2cpx b1;
:: EUCLID_3:th 18
theorem
for b1, b2 being Element of COMPLEX holds
cpx2euc (b1 - b2) = (cpx2euc b1) - cpx2euc b2;
:: EUCLID_3:th 19
theorem
for b1, b2 being Element of the carrier of TOP-REAL 2 holds
euc2cpx (b1 - b2) = (euc2cpx b1) - euc2cpx b2;
:: EUCLID_3:th 20
theorem
cpx2euc 0c = 0.REAL 2;
:: EUCLID_3:th 21
theorem
euc2cpx 0.REAL 2 = 0c;
:: EUCLID_3:th 22
theorem
for b1 being Element of the carrier of TOP-REAL 2
st euc2cpx b1 = 0c
holds b1 = 0.REAL 2;
:: EUCLID_3:th 23
theorem
for b1 being Element of COMPLEX
for b2 being Element of REAL holds
cpx2euc ([*b2,0*] * b1) = b2 * cpx2euc b1;
:: EUCLID_3:th 24
theorem
for b1, b2, b3 being Element of REAL holds
(b1 + (0 * <i>)) * (b2 + (b3 * <i>)) = (b1 * b2) + ((b1 * b3) * <i>);
:: EUCLID_3:th 25
theorem
for b1 being Element of REAL
for b2 being Element of the carrier of TOP-REAL 2 holds
euc2cpx (b1 * b2) = [*b1,0*] * euc2cpx b2;
:: EUCLID_3:th 26
theorem
for b1 being Element of the carrier of TOP-REAL 2 holds
|.euc2cpx b1.| = sqrt (b1 `1 ^2 + (b1 `2 ^2));
:: EUCLID_3:th 27
theorem
for b1 being FinSequence of REAL
st len b1 = 2
holds |.b1.| = sqrt ((b1 . 1) ^2 + ((b1 . 2) ^2));
:: EUCLID_3:th 28
theorem
for b1 being FinSequence of REAL
for b2 being Element of the carrier of TOP-REAL 2
st len b1 = 2 & b2 = b1
holds |.b2.| = |.b1.|;
:: EUCLID_3:th 29
theorem
for b1 being Element of COMPLEX holds
|.cpx2euc b1.| = sqrt ((Re b1) ^2 + ((Im b1) ^2));
:: EUCLID_3:th 30
theorem
for b1 being Element of COMPLEX holds
|.cpx2euc b1.| = |.b1.|;
:: EUCLID_3:th 31
theorem
for b1 being Element of the carrier of TOP-REAL 2 holds
|.euc2cpx b1.| = |.b1.|;
:: EUCLID_3:funcnot 3 => EUCLID_3:func 3
definition
let a1 be Element of the carrier of TOP-REAL 2;
func Arg A1 -> Element of REAL equals
Arg euc2cpx a1;
end;
:: EUCLID_3:def 3
theorem
for b1 being Element of the carrier of TOP-REAL 2 holds
Arg b1 = Arg euc2cpx b1;
:: EUCLID_3:th 32
theorem
for b1 being Element of COMPLEX
for b2 being Element of the carrier of TOP-REAL 2
st (b1 = euc2cpx b2 or b2 = cpx2euc b1)
holds Arg b1 = Arg b2;
:: EUCLID_3:th 34
theorem
for b1, b2 being Element of REAL
for b3 being Element of the carrier of TOP-REAL 2
st b1 = |.b3.| * cos Arg b3 &
b2 = |.b3.| * sin Arg b3
holds b3 = |[b1,b2]|;
:: EUCLID_3:th 35
theorem
Arg 0.REAL 2 = 0;
:: EUCLID_3:th 36
theorem
for b1 being Element of the carrier of TOP-REAL 2
st b1 <> 0.REAL 2
holds (PI <= Arg b1 or Arg - b1 = (Arg b1) + PI) &
(PI <= Arg b1 implies Arg - b1 = (Arg b1) - PI);
:: EUCLID_3:th 37
theorem
for b1 being Element of the carrier of TOP-REAL 2
st Arg b1 = 0
holds b1 = |[|.b1.|,0]| & b1 `2 = 0;
:: EUCLID_3:th 38
theorem
for b1 being Element of the carrier of TOP-REAL 2
st b1 <> 0.REAL 2
holds Arg b1 < PI
iff
PI <= Arg - b1;
:: EUCLID_3:th 39
theorem
for b1, b2 being Element of the carrier of TOP-REAL 2
st (b1 = b2 implies b1 - b2 <> 0.REAL 2)
holds Arg (b1 - b2) < PI
iff
PI <= Arg (b2 - b1);
:: EUCLID_3:th 40
theorem
for b1 being Element of the carrier of TOP-REAL 2 holds
Arg b1 in ].0,PI.[
iff
0 < b1 `2;
:: EUCLID_3:th 42
theorem
for b1, b2 being Element of the carrier of TOP-REAL 2
st Arg b1 < PI & Arg b2 < PI
holds Arg (b1 + b2) < PI;
:: EUCLID_3:funcnot 4 => EUCLID_3:func 4
definition
let a1, a2, a3 be Element of the carrier of TOP-REAL 2;
func angle(A1,A2,A3) -> Element of REAL equals
angle(euc2cpx a1,euc2cpx a2,euc2cpx a3);
end;
:: EUCLID_3:def 4
theorem
for b1, b2, b3 being Element of the carrier of TOP-REAL 2 holds
angle(b1,b2,b3) = angle(euc2cpx b1,euc2cpx b2,euc2cpx b3);
:: EUCLID_3:th 44
theorem
for b1, b2, b3 being Element of the carrier of TOP-REAL 2 holds
angle(b1,b2,b3) = angle(b1 - b2,0.REAL 2,b3 - b2);
:: EUCLID_3:th 45
theorem
for b1, b2, b3 being Element of the carrier of TOP-REAL 2
st angle(b1,b2,b3) = 0
holds Arg (b1 - b2) = Arg (b3 - b2) & angle(b3,b2,b1) = 0;
:: EUCLID_3:th 46
theorem
for b1, b2, b3 being Element of the carrier of TOP-REAL 2
st angle(b1,b2,b3) <> 0
holds angle(b3,b2,b1) = (2 * PI) - angle(b1,b2,b3);
:: EUCLID_3:th 47
theorem
for b1, b2, b3 being Element of the carrier of TOP-REAL 2
st angle(b3,b2,b1) <> 0
holds angle(b3,b2,b1) = (2 * PI) - angle(b1,b2,b3);
:: EUCLID_3:th 48
theorem
for b1, b2 being Element of COMPLEX holds
Re (b1 .|. b2) = ((Re b1) * Re b2) + ((Im b1) * Im b2);
:: EUCLID_3:th 49
theorem
for b1, b2 being Element of COMPLEX holds
Im (b1 .|. b2) = (- ((Re b1) * Im b2)) + ((Im b1) * Re b2);
:: EUCLID_3:th 50
theorem
for b1, b2 being Element of the carrier of TOP-REAL 2 holds
|(b1,b2)| = (b1 `1 * (b2 `1)) + (b1 `2 * (b2 `2));
:: EUCLID_3:th 51
theorem
for b1, b2 being Element of the carrier of TOP-REAL 2 holds
|(b1,b2)| = Re ((euc2cpx b1) .|. euc2cpx b2);
:: EUCLID_3:th 52
theorem
for b1, b2, b3 being Element of the carrier of TOP-REAL 2
st b1 <> 0.REAL 2 & b2 <> 0.REAL 2
holds |(b1,b2)| = 0
iff
(angle(b1,0.REAL 2,b2) = PI / 2 or angle(b1,0.REAL 2,b2) = (3 / 2) * PI);
:: EUCLID_3:th 53
theorem
for b1, b2 being Element of the carrier of TOP-REAL 2
st b1 <> 0.REAL 2 & b2 <> 0.REAL 2
holds ((- (b1 `1 * (b2 `2))) + (b1 `2 * (b2 `1)) = |.b1.| * |.b2.| or (- (b1 `1 * (b2 `2))) + (b1 `2 * (b2 `1)) = - (|.b1.| * |.b2.|))
iff
(angle(b1,0.REAL 2,b2) = PI / 2 or angle(b1,0.REAL 2,b2) = (3 / 2) * PI);
:: EUCLID_3:th 54
theorem
for b1, b2, b3 being Element of the carrier of TOP-REAL 2
st b1 <> b2 & b3 <> b2
holds |(b1 - b2,b3 - b2)| = 0
iff
(angle(b1,b2,b3) = PI / 2 or angle(b1,b2,b3) = (3 / 2) * PI);
:: EUCLID_3:th 55
theorem
for b1, b2, b3 being Element of the carrier of TOP-REAL 2
st b1 <> b2 &
b3 <> b2 &
(angle(b1,b2,b3) = PI / 2 or angle(b1,b2,b3) = (3 / 2) * PI)
holds |.b1 - b2.| ^2 + (|.b3 - b2.| ^2) = |.b1 - b3.| ^2;
:: EUCLID_3:th 56
theorem
for b1, b2, b3 being Element of the carrier of TOP-REAL 2
st b2 <> b1 & b1 <> b3 & b3 <> b2 & angle(b2,b1,b3) < PI & angle(b1,b3,b2) < PI & angle(b3,b2,b1) < PI
holds ((angle(b2,b1,b3)) + angle(b1,b3,b2)) + angle(b3,b2,b1) = PI;
:: EUCLID_3:funcnot 5 => EUCLID_3:func 5
definition
let a1 be Element of NAT;
let a2, a3, a4 be Element of the carrier of TOP-REAL a1;
func Triangle(A2,A3,A4) -> Element of bool the carrier of TOP-REAL a1 equals
((LSeg(a2,a3)) \/ LSeg(a3,a4)) \/ LSeg(a4,a2);
end;
:: EUCLID_3:def 5
theorem
for b1 being Element of NAT
for b2, b3, b4 being Element of the carrier of TOP-REAL b1 holds
Triangle(b2,b3,b4) = ((LSeg(b2,b3)) \/ LSeg(b3,b4)) \/ LSeg(b4,b2);
:: EUCLID_3:funcnot 6 => EUCLID_3:func 6
definition
let a1 be Element of NAT;
let a2, a3, a4 be Element of the carrier of TOP-REAL a1;
func closed_inside_of_triangle(A2,A3,A4) -> Element of bool the carrier of TOP-REAL a1 equals
{b1 where b1 is Element of the carrier of TOP-REAL a1: ex b2, b3, b4 being Element of REAL st
0 <= b2 &
0 <= b3 &
0 <= b4 &
(b2 + b3) + b4 = 1 &
b1 = ((b2 * a2) + (b3 * a3)) + (b4 * a4)};
end;
:: EUCLID_3:def 6
theorem
for b1 being Element of NAT
for b2, b3, b4 being Element of the carrier of TOP-REAL b1 holds
closed_inside_of_triangle(b2,b3,b4) = {b5 where b5 is Element of the carrier of TOP-REAL b1: ex b6, b7, b8 being Element of REAL st
0 <= b6 &
0 <= b7 &
0 <= b8 &
(b6 + b7) + b8 = 1 &
b5 = ((b6 * b2) + (b7 * b3)) + (b8 * b4)};
:: EUCLID_3:funcnot 7 => EUCLID_3:func 7
definition
let a1 be Element of NAT;
let a2, a3, a4 be Element of the carrier of TOP-REAL a1;
func inside_of_triangle(A2,A3,A4) -> Element of bool the carrier of TOP-REAL a1 equals
(closed_inside_of_triangle(a2,a3,a4)) \ Triangle(a2,a3,a4);
end;
:: EUCLID_3:def 7
theorem
for b1 being Element of NAT
for b2, b3, b4 being Element of the carrier of TOP-REAL b1 holds
inside_of_triangle(b2,b3,b4) = (closed_inside_of_triangle(b2,b3,b4)) \ Triangle(b2,b3,b4);
:: EUCLID_3:funcnot 8 => EUCLID_3:func 8
definition
let a1 be Element of NAT;
let a2, a3, a4 be Element of the carrier of TOP-REAL a1;
func outside_of_triangle(A2,A3,A4) -> Element of bool the carrier of TOP-REAL a1 equals
{b1 where b1 is Element of the carrier of TOP-REAL a1: ex b2, b3, b4 being Element of REAL st
(0 <= b2 & 0 <= b3 implies b4 < 0) &
(b2 + b3) + b4 = 1 &
b1 = ((b2 * a2) + (b3 * a3)) + (b4 * a4)};
end;
:: EUCLID_3:def 8
theorem
for b1 being Element of NAT
for b2, b3, b4 being Element of the carrier of TOP-REAL b1 holds
outside_of_triangle(b2,b3,b4) = {b5 where b5 is Element of the carrier of TOP-REAL b1: ex b6, b7, b8 being Element of REAL st
(0 <= b6 & 0 <= b7 implies b8 < 0) &
(b6 + b7) + b8 = 1 &
b5 = ((b6 * b2) + (b7 * b3)) + (b8 * b4)};
:: EUCLID_3:funcnot 9 => EUCLID_3:func 9
definition
let a1 be Element of NAT;
let a2, a3, a4 be Element of the carrier of TOP-REAL a1;
func plane(A2,A3,A4) -> Element of bool the carrier of TOP-REAL a1 equals
(outside_of_triangle(a2,a3,a4)) \/ closed_inside_of_triangle(a2,a3,a4);
end;
:: EUCLID_3:def 9
theorem
for b1 being Element of NAT
for b2, b3, b4 being Element of the carrier of TOP-REAL b1 holds
plane(b2,b3,b4) = (outside_of_triangle(b2,b3,b4)) \/ closed_inside_of_triangle(b2,b3,b4);
:: EUCLID_3:th 57
theorem
for b1 being Element of NAT
for b2, b3, b4, b5 being Element of the carrier of TOP-REAL b1
st b5 in plane(b2,b3,b4)
holds ex b6, b7, b8 being Element of REAL st
(b6 + b7) + b8 = 1 &
b5 = ((b6 * b2) + (b7 * b3)) + (b8 * b4);
:: EUCLID_3:th 58
theorem
for b1 being Element of NAT
for b2, b3, b4 being Element of the carrier of TOP-REAL b1 holds
Triangle(b2,b3,b4) c= closed_inside_of_triangle(b2,b3,b4);
:: EUCLID_3:prednot 1 => EUCLID_3:pred 1
definition
let a1 be Element of NAT;
let a2, a3 be Element of the carrier of TOP-REAL a1;
pred A2,A3 are_lindependent2 means
for b1, b2 being Element of REAL
st (b1 * a2) + (b2 * a3) = 0.REAL a1
holds b1 = 0 & b2 = 0;
end;
:: EUCLID_3:dfs 10
definiens
let a1 be Element of NAT;
let a2, a3 be Element of the carrier of TOP-REAL a1;
To prove
a2,a3 are_lindependent2
it is sufficient to prove
thus for b1, b2 being Element of REAL
st (b1 * a2) + (b2 * a3) = 0.REAL a1
holds b1 = 0 & b2 = 0;
:: EUCLID_3:def 10
theorem
for b1 being Element of NAT
for b2, b3 being Element of the carrier of TOP-REAL b1 holds
b2,b3 are_lindependent2
iff
for b4, b5 being Element of REAL
st (b4 * b2) + (b5 * b3) = 0.REAL b1
holds b4 = 0 & b5 = 0;
:: EUCLID_3:prednot 2 => not EUCLID_3:pred 1
notation
let a1 be Element of NAT;
let a2, a3 be Element of the carrier of TOP-REAL a1;
antonym a2,a3 are_ldependent2 for a2,a3 are_lindependent2;
end;
:: EUCLID_3:th 59
theorem
for b1 being Element of NAT
for b2, b3 being Element of the carrier of TOP-REAL b1
st b2,b3 are_lindependent2
holds b2 <> b3 & b2 <> 0.REAL b1 & b3 <> 0.REAL b1;
:: EUCLID_3:th 60
theorem
for b1 being Element of NAT
for b2, b3, b4, b5 being Element of the carrier of TOP-REAL b1
st b3 - b2,b4 - b2 are_lindependent2 & b5 in plane(b2,b3,b4)
holds ex b6, b7, b8 being Element of REAL st
b5 = ((b6 * b2) + (b7 * b3)) + (b8 * b4) &
(b6 + b7) + b8 = 1 &
(for b9, b10, b11 being Element of REAL
st b5 = ((b9 * b2) + (b10 * b3)) + (b11 * b4) &
(b9 + b10) + b11 = 1
holds b9 = b6 & b10 = b7 & b11 = b8);
:: EUCLID_3:th 61
theorem
for b1 being Element of NAT
for b2, b3, b4, b5 being Element of the carrier of TOP-REAL b1
st ex b6, b7, b8 being Element of REAL st
b5 = ((b6 * b2) + (b7 * b3)) + (b8 * b4) &
(b6 + b7) + b8 = 1
holds b5 in plane(b2,b3,b4);
:: EUCLID_3:th 62
theorem
for b1 being Element of NAT
for b2, b3, b4 being Element of the carrier of TOP-REAL b1 holds
plane(b2,b3,b4) = {b5 where b5 is Element of the carrier of TOP-REAL b1: ex b6, b7, b8 being Element of REAL st
(b6 + b7) + b8 = 1 &
b5 = ((b6 * b2) + (b7 * b3)) + (b8 * b4)};
:: EUCLID_3:th 63
theorem
for b1, b2, b3 being Element of the carrier of TOP-REAL 2
st b2 - b1,b3 - b1 are_lindependent2
holds plane(b1,b2,b3) = REAL 2;
:: EUCLID_3:funcnot 10 => EUCLID_3:func 10
definition
let a1 be Element of NAT;
let a2, a3, a4, a5 be Element of the carrier of TOP-REAL a1;
assume a3 - a2,a4 - a2 are_lindependent2 & a5 in plane(a2,a3,a4);
func tricord1(A2,A3,A4,A5) -> Element of REAL means
ex b1, b2 being Element of REAL st
(it + b1) + b2 = 1 &
a5 = ((it * a2) + (b1 * a3)) + (b2 * a4);
end;
:: EUCLID_3:def 11
theorem
for b1 being Element of NAT
for b2, b3, b4, b5 being Element of the carrier of TOP-REAL b1
st b3 - b2,b4 - b2 are_lindependent2 & b5 in plane(b2,b3,b4)
for b6 being Element of REAL holds
b6 = tricord1(b2,b3,b4,b5)
iff
ex b7, b8 being Element of REAL st
(b6 + b7) + b8 = 1 &
b5 = ((b6 * b2) + (b7 * b3)) + (b8 * b4);
:: EUCLID_3:funcnot 11 => EUCLID_3:func 11
definition
let a1 be Element of NAT;
let a2, a3, a4, a5 be Element of the carrier of TOP-REAL a1;
assume a3 - a2,a4 - a2 are_lindependent2 & a5 in plane(a2,a3,a4);
func tricord2(A2,A3,A4,A5) -> Element of REAL means
ex b1, b2 being Element of REAL st
(b1 + it) + b2 = 1 &
a5 = ((b1 * a2) + (it * a3)) + (b2 * a4);
end;
:: EUCLID_3:def 12
theorem
for b1 being Element of NAT
for b2, b3, b4, b5 being Element of the carrier of TOP-REAL b1
st b3 - b2,b4 - b2 are_lindependent2 & b5 in plane(b2,b3,b4)
for b6 being Element of REAL holds
b6 = tricord2(b2,b3,b4,b5)
iff
ex b7, b8 being Element of REAL st
(b7 + b6) + b8 = 1 &
b5 = ((b7 * b2) + (b6 * b3)) + (b8 * b4);
:: EUCLID_3:funcnot 12 => EUCLID_3:func 12
definition
let a1 be Element of NAT;
let a2, a3, a4, a5 be Element of the carrier of TOP-REAL a1;
assume a3 - a2,a4 - a2 are_lindependent2 & a5 in plane(a2,a3,a4);
func tricord3(A2,A3,A4,A5) -> Element of REAL means
ex b1, b2 being Element of REAL st
(b1 + b2) + it = 1 &
a5 = ((b1 * a2) + (b2 * a3)) + (it * a4);
end;
:: EUCLID_3:def 13
theorem
for b1 being Element of NAT
for b2, b3, b4, b5 being Element of the carrier of TOP-REAL b1
st b3 - b2,b4 - b2 are_lindependent2 & b5 in plane(b2,b3,b4)
for b6 being Element of REAL holds
b6 = tricord3(b2,b3,b4,b5)
iff
ex b7, b8 being Element of REAL st
(b7 + b8) + b6 = 1 &
b5 = ((b7 * b2) + (b8 * b3)) + (b6 * b4);
:: EUCLID_3:funcnot 13 => EUCLID_3:func 13
definition
let a1, a2, a3 be Element of the carrier of TOP-REAL 2;
func trcmap1(A1,A2,A3) -> Function-like quasi_total Relation of the carrier of TOP-REAL 2,the carrier of R^1 means
for b1 being Element of the carrier of TOP-REAL 2 holds
it . b1 = tricord1(a1,a2,a3,b1);
end;
:: EUCLID_3:def 14
theorem
for b1, b2, b3 being Element of the carrier of TOP-REAL 2
for b4 being Function-like quasi_total Relation of the carrier of TOP-REAL 2,the carrier of R^1 holds
b4 = trcmap1(b1,b2,b3)
iff
for b5 being Element of the carrier of TOP-REAL 2 holds
b4 . b5 = tricord1(b1,b2,b3,b5);
:: EUCLID_3:funcnot 14 => EUCLID_3:func 14
definition
let a1, a2, a3 be Element of the carrier of TOP-REAL 2;
func trcmap2(A1,A2,A3) -> Function-like quasi_total Relation of the carrier of TOP-REAL 2,the carrier of R^1 means
for b1 being Element of the carrier of TOP-REAL 2 holds
it . b1 = tricord2(a1,a2,a3,b1);
end;
:: EUCLID_3:def 15
theorem
for b1, b2, b3 being Element of the carrier of TOP-REAL 2
for b4 being Function-like quasi_total Relation of the carrier of TOP-REAL 2,the carrier of R^1 holds
b4 = trcmap2(b1,b2,b3)
iff
for b5 being Element of the carrier of TOP-REAL 2 holds
b4 . b5 = tricord2(b1,b2,b3,b5);
:: EUCLID_3:funcnot 15 => EUCLID_3:func 15
definition
let a1, a2, a3 be Element of the carrier of TOP-REAL 2;
func trcmap3(A1,A2,A3) -> Function-like quasi_total Relation of the carrier of TOP-REAL 2,the carrier of R^1 means
for b1 being Element of the carrier of TOP-REAL 2 holds
it . b1 = tricord3(a1,a2,a3,b1);
end;
:: EUCLID_3:def 16
theorem
for b1, b2, b3 being Element of the carrier of TOP-REAL 2
for b4 being Function-like quasi_total Relation of the carrier of TOP-REAL 2,the carrier of R^1 holds
b4 = trcmap3(b1,b2,b3)
iff
for b5 being Element of the carrier of TOP-REAL 2 holds
b4 . b5 = tricord3(b1,b2,b3,b5);
:: EUCLID_3:th 64
theorem
for b1, b2, b3, b4 being Element of the carrier of TOP-REAL 2
st b2 - b1,b3 - b1 are_lindependent2
holds b4 in outside_of_triangle(b1,b2,b3)
iff
(0 <= tricord1(b1,b2,b3,b4) & 0 <= tricord2(b1,b2,b3,b4) implies tricord3(b1,b2,b3,b4) < 0);
:: EUCLID_3:th 65
theorem
for b1, b2, b3, b4 being Element of the carrier of TOP-REAL 2
st b2 - b1,b3 - b1 are_lindependent2
holds b4 in Triangle(b1,b2,b3)
iff
0 <= tricord1(b1,b2,b3,b4) &
0 <= tricord2(b1,b2,b3,b4) &
0 <= tricord3(b1,b2,b3,b4) &
(tricord1(b1,b2,b3,b4) <> 0 & tricord2(b1,b2,b3,b4) <> 0 implies tricord3(b1,b2,b3,b4) = 0);
:: EUCLID_3:th 66
theorem
for b1, b2, b3, b4 being Element of the carrier of TOP-REAL 2
st b2 - b1,b3 - b1 are_lindependent2
holds b4 in Triangle(b1,b2,b3)
iff
((tricord1(b1,b2,b3,b4) = 0 & 0 <= tricord2(b1,b2,b3,b4) implies tricord3(b1,b2,b3,b4) < 0) &
(0 <= tricord1(b1,b2,b3,b4) & tricord2(b1,b2,b3,b4) = 0 implies tricord3(b1,b2,b3,b4) < 0) implies 0 <= tricord1(b1,b2,b3,b4) & 0 <= tricord2(b1,b2,b3,b4) & tricord3(b1,b2,b3,b4) = 0);
:: EUCLID_3:th 67
theorem
for b1, b2, b3, b4 being Element of the carrier of TOP-REAL 2
st b2 - b1,b3 - b1 are_lindependent2
holds b4 in inside_of_triangle(b1,b2,b3)
iff
0 < tricord1(b1,b2,b3,b4) & 0 < tricord2(b1,b2,b3,b4) & 0 < tricord3(b1,b2,b3,b4);
:: EUCLID_3:th 68
theorem
for b1, b2, b3 being Element of the carrier of TOP-REAL 2
st b2 - b1,b3 - b1 are_lindependent2
holds inside_of_triangle(b1,b2,b3) is not empty;