Article SQUARE_1, MML version 4.99.1005
:: SQUARE_1:th 2
theorem
for b1 being real set
st 1 < b1
holds 1 / b1 < 1;
:: SQUARE_1:th 25
theorem
for b1, b2 being real set
st 0 <= b1 * b2 & (0 <= b1 implies b2 < 0)
holds b1 <= 0 & b2 <= 0;
:: SQUARE_1:sch 1
scheme SQUARE_1:sch 1
ex b1 being real set st
for b2, b3 being real set
st P1[b2] & P2[b3]
holds b2 <= b1 & b1 <= b3
provided
for b1, b2 being real set
st P1[b1] & P2[b2]
holds b1 <= b2;
:: SQUARE_1:funcnot 1 => SQUARE_1:func 1
definition
let a1, a2 be Element of REAL;
redefine func min(a1,a2) -> Element of REAL;
commutativity;
:: for a1, a2 being Element of REAL holds
:: min(a1,a2) = min(a2,a1);
idempotence;
:: for a1 being Element of REAL holds
:: min(a1,a1) = a1;
end;
:: SQUARE_1:funcnot 2 => SQUARE_1:func 2
definition
let a1, a2 be Element of REAL;
redefine func max(a1,a2) -> Element of REAL;
commutativity;
:: for a1, a2 being Element of REAL holds
:: max(a1,a2) = max(a2,a1);
idempotence;
:: for a1 being Element of REAL holds
:: max(a1,a1) = a1;
end;
:: SQUARE_1:th 39
theorem
for b1, b2, b3 being real set holds
b1 <= b2 & b1 <= b3
iff
b1 <= min(b2,b3);
:: SQUARE_1:th 50
theorem
for b1, b2, b3 being real set holds
b1 <= b2 & b3 <= b2
iff
max(b1,b3) <= b2;
:: SQUARE_1:th 53
theorem
for b1, b2 being real set holds
(min(b1,b2)) + max(b1,b2) = b1 + b2;
:: SQUARE_1:funcnot 3 => SQUARE_1:func 3
definition
let a1 be complex set;
func A1 ^2 -> set equals
a1 * a1;
end;
:: SQUARE_1:def 3
theorem
for b1 being complex set holds
b1 ^2 = b1 * b1;
:: SQUARE_1:funcreg 1
registration
let a1 be complex set;
cluster a1 ^2 -> complex;
end;
:: SQUARE_1:funcreg 2
registration
let a1 be real set;
cluster a1 ^2 -> real;
end;
:: SQUARE_1:funcnot 4 => SQUARE_1:func 4
definition
let a1 be Element of COMPLEX;
redefine func a1 ^2 -> Element of COMPLEX;
end;
:: SQUARE_1:funcnot 5 => SQUARE_1:func 5
definition
let a1 be Element of REAL;
redefine func a1 ^2 -> Element of REAL;
end;
:: SQUARE_1:th 59
theorem
1 ^2 = 1;
:: SQUARE_1:th 60
theorem
0 ^2 = 0;
:: SQUARE_1:th 61
theorem
for b1 being complex set holds
b1 ^2 = (- b1) ^2;
:: SQUARE_1:th 63
theorem
for b1, b2 being complex set holds
(b1 + b2) ^2 = (b1 ^2 + ((2 * b1) * b2)) + (b2 ^2);
:: SQUARE_1:th 64
theorem
for b1, b2 being complex set holds
(b1 - b2) ^2 = (b1 ^2 - ((2 * b1) * b2)) + (b2 ^2);
:: SQUARE_1:th 65
theorem
for b1 being complex set holds
(b1 + 1) ^2 = (b1 ^2 + (2 * b1)) + 1;
:: SQUARE_1:th 66
theorem
for b1 being complex set holds
(b1 - 1) ^2 = (b1 ^2 - (2 * b1)) + 1;
:: SQUARE_1:th 67
theorem
for b1, b2 being complex set holds
(b1 - b2) * (b1 + b2) = b1 ^2 - (b2 ^2);
:: SQUARE_1:th 68
theorem
for b1, b2 being complex set holds
(b1 * b2) ^2 = b1 ^2 * (b2 ^2);
:: SQUARE_1:th 70
theorem
for b1, b2 being complex set
st b1 ^2 - (b2 ^2) <> 0
holds 1 / (b1 + b2) = (b1 - b2) / (b1 ^2 - (b2 ^2));
:: SQUARE_1:th 71
theorem
for b1, b2 being complex set
st b1 ^2 - (b2 ^2) <> 0
holds 1 / (b1 - b2) = (b1 + b2) / (b1 ^2 - (b2 ^2));
:: SQUARE_1:th 74
theorem
for b1 being real set
st 0 <> b1
holds 0 < b1 ^2;
:: SQUARE_1:th 75
theorem
for b1 being real set
st 0 < b1 & b1 < 1
holds b1 ^2 < b1;
:: SQUARE_1:th 76
theorem
for b1 being real set
st 1 < b1
holds b1 < b1 ^2;
:: SQUARE_1:th 77
theorem
for b1, b2 being real set
st 0 <= b1 & b1 <= b2
holds b1 ^2 <= b2 ^2;
:: SQUARE_1:th 78
theorem
for b1, b2 being real set
st 0 <= b1 & b1 < b2
holds b1 ^2 < b2 ^2;
:: SQUARE_1:funcnot 6 => SQUARE_1:func 6
definition
let a1 be real set;
assume 0 <= a1;
func sqrt A1 -> real set means
0 <= it & it ^2 = a1;
end;
:: SQUARE_1:def 4
theorem
for b1 being real set
st 0 <= b1
for b2 being real set holds
b2 = sqrt b1
iff
0 <= b2 & b2 ^2 = b1;
:: SQUARE_1:funcnot 7 => SQUARE_1:func 7
definition
let a1 be Element of REAL;
redefine func sqrt a1 -> Element of REAL;
end;
:: SQUARE_1:th 82
theorem
sqrt 0 = 0;
:: SQUARE_1:th 83
theorem
sqrt 1 = 1;
:: SQUARE_1:th 84
theorem
1 < sqrt 2;
:: SQUARE_1:th 85
theorem
sqrt 4 = 2;
:: SQUARE_1:th 86
theorem
sqrt 2 < 2;
:: SQUARE_1:th 89
theorem
for b1 being real set
st 0 <= b1
holds sqrt (b1 ^2) = b1;
:: SQUARE_1:th 90
theorem
for b1 being real set
st b1 <= 0
holds sqrt (b1 ^2) = - b1;
:: SQUARE_1:th 92
theorem
for b1 being real set
st 0 <= b1 & sqrt b1 = 0
holds b1 = 0;
:: SQUARE_1:th 93
theorem
for b1 being real set
st 0 < b1
holds 0 < sqrt b1;
:: SQUARE_1:th 94
theorem
for b1, b2 being real set
st 0 <= b1 & b1 <= b2
holds sqrt b1 <= sqrt b2;
:: SQUARE_1:th 95
theorem
for b1, b2 being real set
st 0 <= b1 & b1 < b2
holds sqrt b1 < sqrt b2;
:: SQUARE_1:th 96
theorem
for b1, b2 being real set
st 0 <= b1 & 0 <= b2 & sqrt b1 = sqrt b2
holds b1 = b2;
:: SQUARE_1:th 97
theorem
for b1, b2 being real set
st 0 <= b1 & 0 <= b2
holds sqrt (b1 * b2) = (sqrt b1) * sqrt b2;
:: SQUARE_1:th 99
theorem
for b1, b2 being real set
st 0 <= b1 & 0 <= b2
holds sqrt (b1 / b2) = (sqrt b1) / sqrt b2;
:: SQUARE_1:th 101
theorem
for b1 being real set
st 0 < b1
holds sqrt (1 / b1) = 1 / sqrt b1;
:: SQUARE_1:th 102
theorem
for b1 being real set
st 0 < b1
holds (sqrt b1) / b1 = 1 / sqrt b1;
:: SQUARE_1:th 103
theorem
for b1 being real set
st 0 < b1
holds b1 / sqrt b1 = sqrt b1;
:: SQUARE_1:th 104
theorem
for b1, b2 being real set
st 0 <= b1 & 0 <= b2
holds ((sqrt b1) - sqrt b2) * ((sqrt b1) + sqrt b2) = b1 - b2;
:: SQUARE_1:th 105
theorem
for b1, b2 being real set
st 0 <= b1 & 0 <= b2 & b1 <> b2
holds 1 / ((sqrt b1) + sqrt b2) = ((sqrt b1) - sqrt b2) / (b1 - b2);
:: SQUARE_1:th 106
theorem
for b1, b2 being real set
st 0 <= b1 & b1 < b2
holds 1 / ((sqrt b2) + sqrt b1) = ((sqrt b2) - sqrt b1) / (b2 - b1);
:: SQUARE_1:th 107
theorem
for b1, b2 being real set
st 0 <= b1 & 0 <= b2
holds 1 / ((sqrt b1) - sqrt b2) = ((sqrt b1) + sqrt b2) / (b1 - b2);
:: SQUARE_1:th 108
theorem
for b1, b2 being real set
st 0 <= b1 & b1 < b2
holds 1 / ((sqrt b2) - sqrt b1) = ((sqrt b2) + sqrt b1) / (b2 - b1);
:: SQUARE_1:th 109
theorem
for b1, b2 being complex set
st b1 ^2 = b2 ^2 & b1 <> b2
holds b1 = - b2;
:: SQUARE_1:th 110
theorem
for b1 being complex set
st b1 ^2 = 1 & b1 <> 1
holds b1 = - 1;
:: SQUARE_1:th 111
theorem
for b1 being real set
st 0 <= b1 & b1 <= 1
holds b1 ^2 <= b1;
:: SQUARE_1:th 112
theorem
for b1 being real set
st b1 ^2 - 1 <= 0
holds - 1 <= b1 & b1 <= 1;
:: SQUARE_1:th 113
theorem
for b1, b2, b3 being real set holds
b1 < b2 & b1 < b3
iff
b1 < min(b2,b3);
:: SQUARE_1:th 114
theorem
for b1, b2 being real set
st b1 <= 0 & b2 < b1
holds b1 ^2 < b2 ^2;
:: SQUARE_1:th 115
theorem
for b1 being real set
st b1 <= - 1
holds - b1 <= b1 ^2;
:: SQUARE_1:th 116
theorem
for b1 being real set
st b1 < - 1
holds - b1 < b1 ^2;
:: SQUARE_1:th 117
theorem
for b1, b2 being real set
st b1 ^2 <= b2 ^2 & 0 <= b2
holds - b2 <= b1 & b1 <= b2;
:: SQUARE_1:th 118
theorem
for b1, b2 being real set
st b1 ^2 < b2 ^2 & 0 <= b2
holds - b2 < b1 & b1 < b2;
:: SQUARE_1:th 119
theorem
for b1, b2 being real set
st - b1 <= b2 & b2 <= b1
holds b2 ^2 <= b1 ^2;
:: SQUARE_1:th 120
theorem
for b1, b2 being real set
st - b1 < b2 & b2 < b1
holds b2 ^2 < b1 ^2;
:: SQUARE_1:th 121
theorem
for b1 being real set
st b1 ^2 <= 1
holds - 1 <= b1 & b1 <= 1;
:: SQUARE_1:th 122
theorem
for b1 being real set
st b1 ^2 < 1
holds - 1 < b1 & b1 < 1;
:: SQUARE_1:th 123
theorem
for b1, b2 being real set
st - 1 <= b1 & b1 <= 1 & - 1 <= b2 & b2 <= 1
holds b1 ^2 * (b2 ^2) <= 1;
:: SQUARE_1:th 124
theorem
for b1, b2 being real set
st 0 <= b1 & 0 <= b2
holds b1 * sqrt b2 = sqrt (b1 ^2 * b2);
:: SQUARE_1:th 125
theorem
for b1, b2 being real set
st - 1 <= b1 & b1 <= 1 & - 1 <= b2 & b2 <= 1
holds (- b2) * sqrt (1 + (b1 ^2)) <= sqrt (1 + (b2 ^2)) &
- sqrt (1 + (b2 ^2)) <= b2 * sqrt (1 + (b1 ^2));
:: SQUARE_1:th 126
theorem
for b1, b2 being real set
st - 1 <= b1 & b1 <= 1 & - 1 <= b2 & b2 <= 1
holds b2 * sqrt (1 + (b1 ^2)) <= sqrt (1 + (b2 ^2));
:: SQUARE_1:th 127
theorem
for b1, b2 being real set
st b2 <= b1
holds b2 * sqrt (1 + (b1 ^2)) <= b1 * sqrt (1 + (b2 ^2));