:: The Quaternion Numbers
:: by Xiquan Liang and Fuguo Ge
::
:: Received November 14, 2006
:: Copyright (c) 2006-2012 Association of Mizar Users


begin

definition
func QUATERNION -> set equals :: QUATERNI:def 1
((Funcs (4,REAL)) \ { x where x is Element of Funcs (4,REAL) : ( x . 2 = 0 & x . 3 = 0 ) } ) \/ COMPLEX;
coherence
((Funcs (4,REAL)) \ { x where x is Element of Funcs (4,REAL) : ( x . 2 = 0 & x . 3 = 0 ) } ) \/ COMPLEX is set ;
end;

:: deftheorem defines QUATERNION QUATERNI:def 1 :
QUATERNION = ((Funcs (4,REAL)) \ { x where x is Element of Funcs (4,REAL) : ( x . 2 = 0 & x . 3 = 0 ) } ) \/ COMPLEX;

definition
let x be number ;
attr x is quaternion means :Def2: :: QUATERNI:def 2
x in QUATERNION ;
end;

:: deftheorem Def2 defines quaternion QUATERNI:def 2 :
for x being number holds
( x is quaternion iff x in QUATERNION );

registration
cluster QUATERNION -> non empty ;
coherence
not QUATERNION is empty ;
end;

definition
let x, y, w, z, a, b, c, d be set ;
func (x,y,w,z) --> (a,b,c,d) -> set equals :: QUATERNI:def 3
((x,y) --> (a,b)) +* ((w,z) --> (c,d));
coherence
((x,y) --> (a,b)) +* ((w,z) --> (c,d)) is set ;
end;

:: deftheorem defines --> QUATERNI:def 3 :
for x, y, w, z, a, b, c, d being set holds (x,y,w,z) --> (a,b,c,d) = ((x,y) --> (a,b)) +* ((w,z) --> (c,d));

registration
let x, y, w, z, a, b, c, d be set ;
cluster (x,y,w,z) --> (a,b,c,d) -> Relation-like Function-like ;
coherence
( (x,y,w,z) --> (a,b,c,d) is Function-like & (x,y,w,z) --> (a,b,c,d) is Relation-like ) ;
end;

theorem Th1: :: QUATERNI:1
for x, y, w, z, a, b, c, d being set holds dom ((x,y,w,z) --> (a,b,c,d)) = {x,y,w,z}
proof end;

theorem Th2: :: QUATERNI:2
for x, y, w, z, a, b, c, d being set holds rng ((x,y,w,z) --> (a,b,c,d)) c= {a,b,c,d}
proof end;

Lm1: 0 ,1,2,3 are_mutually_different
by ZFMISC_1:def 6;

theorem Th3: :: QUATERNI:3
for x, y, w, z, a, b, c, d being set st x,y,w,z are_mutually_different holds
( ((x,y,w,z) --> (a,b,c,d)) . x = a & ((x,y,w,z) --> (a,b,c,d)) . y = b & ((x,y,w,z) --> (a,b,c,d)) . w = c & ((x,y,w,z) --> (a,b,c,d)) . z = d )
proof end;

Lm2: for x, y, w, z, a, b, c, d being set st x,y,w,z are_mutually_different holds
{a,b,c,d} c= rng ((x,y,w,z) --> (a,b,c,d))
proof end;

theorem :: QUATERNI:4
for x, y, w, z, a, b, c, d being set st x,y,w,z are_mutually_different holds
rng ((x,y,w,z) --> (a,b,c,d)) = {a,b,c,d}
proof end;

theorem :: QUATERNI:5
for x1, x2, x3, x4, X being set holds
( {x1,x2,x3,x4} c= X iff ( x1 in X & x2 in X & x3 in X & x4 in X ) )
proof end;

definition
let A be non empty set ;
let x, y, w, z be set ;
let a, b, c, d be Element of A;
:: original: -->
redefine func (x,y,w,z) --> (a,b,c,d) -> Function of {x,y,w,z},A;
coherence
(x,y,w,z) --> (a,b,c,d) is Function of {x,y,w,z},A
proof end;
end;

definition
func <j> -> set equals :: QUATERNI:def 4
(0,1,2,3) --> (0,0,1,0);
coherence
(0,1,2,3) --> (0,0,1,0) is set ;
func <k> -> set equals :: QUATERNI:def 5
(0,1,2,3) --> (0,0,0,1);
coherence
(0,1,2,3) --> (0,0,0,1) is set ;
end;

:: deftheorem defines <j> QUATERNI:def 4 :
<j> = (0,1,2,3) --> (0,0,1,0);

:: deftheorem defines <k> QUATERNI:def 5 :
<k> = (0,1,2,3) --> (0,0,0,1);

Lm3: <i> = [*0,1*]
by ARYTM_0:def 5;

registration
cluster <i> -> quaternion ;
coherence
<i> is quaternion
proof end;
cluster <j> -> quaternion ;
coherence
<j> is quaternion
proof end;
cluster <k> -> quaternion ;
coherence
<k> is quaternion
proof end;
end;

registration
cluster quaternion for set ;
existence
ex b1 being number st b1 is quaternion
proof end;
end;

registration
cluster -> quaternion for Element of QUATERNION ;
coherence
for b1 being Element of QUATERNION holds b1 is quaternion by Def2;
end;

definition
let x, y, w, z be real number ;
func [*x,y,w,z*] -> Element of QUATERNION means :Def6: :: QUATERNI:def 6
 ex x9, y9 being Element of REAL st
( x9 = x & y9 = y & it = [*x9,y9*] ) if ( w = 0 & z = 0 )
otherwise it = (0,1,2,3) --> (x,y,w,z);
consistency
for b1 being Element of QUATERNION holds verum ;
existence
( ( w = 0 & z = 0 implies ex b1 being Element of QUATERNION ex x9, y9 being Element of REAL st
( x9 = x & y9 = y & b1 = [*x9,y9*] ) ) & ( ( not w = 0 or not z = 0 ) implies ex b1 being Element of QUATERNION st b1 = (0,1,2,3) --> (x,y,w,z) ) )
proof end;
uniqueness
for b1, b2 being Element of QUATERNION holds
( ( w = 0 & z = 0 & ex x9, y9 being Element of REAL st
( x9 = x & y9 = y & b1 = [*x9,y9*] ) & ex x9, y9 being Element of REAL st
( x9 = x & y9 = y & b2 = [*x9,y9*] ) implies b1 = b2 ) & ( ( not w = 0 or not z = 0 ) & b1 = (0,1,2,3) --> (x,y,w,z) & b2 = (0,1,2,3) --> (x,y,w,z) implies b1 = b2 ) ) ;
end;

:: deftheorem Def6 defines [* QUATERNI:def 6 :
for x, y, w, z being real number
for b5 being Element of QUATERNION holds
( ( w = 0 & z = 0 implies ( b5 = [*x,y,w,z*] iff ex x9, y9 being Element of REAL st
( x9 = x & y9 = y & b5 = [*x9,y9*] ) ) ) & ( ( not w = 0 or not z = 0 ) implies ( b5 = [*x,y,w,z*] iff b5 = (0,1,2,3) --> (x,y,w,z) ) ) );

Lm4: for x, y being Element of REAL holds [*x,y,0,0*] = [*x,y*]
proof end;

theorem Th6: :: QUATERNI:6
for a, b, c, d, e, i, j, k being set
for g being Function st a <> b & c <> d & dom g = {a,b,c,d} & g . a = e & g . b = i & g . c = j & g . d = k holds
g = (a,b,c,d) --> (e,i,j,k)
proof end;

theorem Th7: :: QUATERNI:7
for g being quaternion number ex r, s, t, u being Element of REAL st g = [*r,s,t,u*]
proof end;

theorem Th8: :: QUATERNI:8
for a, c, x, w, b, d, y, z being set st a,c,x,w are_mutually_different holds
(a,c,x,w) --> (b,d,y,z) = {[a,b],[c,d],[x,y],[w,z]}
proof end;

Lm5: for x, y, z being set st [x,y] = {z} holds
( z = {x} & x = y )
proof end;

theorem Th9: :: QUATERNI:9
for A being Subset of RAT+ st ex t being Element of RAT+ st
( t in A & t <> {} ) & ( for r, s being Element of RAT+ st r in A & s <=' r holds
s in A ) holds
ex r1, r2, r3, r4, r5 being Element of RAT+ st
( r1 in A & r2 in A & r3 in A & r4 in A & r5 in A & r1 <> r2 & r1 <> r3 & r1 <> r4 & r1 <> r5 & r2 <> r3 & r2 <> r4 & r2 <> r5 & r3 <> r4 & r3 <> r5 & r4 <> r5 )
proof end;

Lm6: for a, b, c, d being Element of REAL holds not (0,1,2,3) --> (a,b,c,d) in REAL
proof end;

theorem Th10: :: QUATERNI:10
for a, b, c, d being Element of REAL holds not (0,1,2,3) --> (a,b,c,d) in COMPLEX
proof end;

theorem Th11: :: QUATERNI:11
for a, b, c, d, x, y, z, w, x9, y9, z9, w9 being set st a,b,c,d are_mutually_different & (a,b,c,d) --> (x,y,z,w) = (a,b,c,d) --> (x9,y9,z9,w9) holds
( x = x9 & y = y9 & z = z9 & w = w9 )
proof end;

theorem Th12: :: QUATERNI:12
for x1, x2, x3, x4, y1, y2, y3, y4 being Element of REAL st [*x1,x2,x3,x4*] = [*y1,y2,y3,y4*] holds
( x1 = y1 & x2 = y2 & x3 = y3 & x4 = y4 )
proof end;

definition
let x, y be quaternion number ;
consider x1, x2, x3, x4 being Element of REAL such that
A1: x = [*x1,x2,x3,x4*] by Th7;
consider y1, y2, y3, y4 being Element of REAL such that
A2: y = [*y1,y2,y3,y4*] by Th7;
func x + y -> set means :Def7: :: QUATERNI:def 7
 ex x1, x2, x3, x4, y1, y2, y3, y4 being Element of REAL st
( x = [*x1,x2,x3,x4*] & y = [*y1,y2,y3,y4*] & it = [*(x1 + y1),(x2 + y2),(x3 + y3),(x4 + y4)*] );
existence
ex b1 being set ex x1, x2, x3, x4, y1, y2, y3, y4 being Element of REAL st
( x = [*x1,x2,x3,x4*] & y = [*y1,y2,y3,y4*] & b1 = [*(x1 + y1),(x2 + y2),(x3 + y3),(x4 + y4)*] )
proof end;
uniqueness
for b1, b2 being set st ex x1, x2, x3, x4, y1, y2, y3, y4 being Element of REAL st
( x = [*x1,x2,x3,x4*] & y = [*y1,y2,y3,y4*] & b1 = [*(x1 + y1),(x2 + y2),(x3 + y3),(x4 + y4)*] ) & ex x1, x2, x3, x4, y1, y2, y3, y4 being Element of REAL st
( x = [*x1,x2,x3,x4*] & y = [*y1,y2,y3,y4*] & b2 = [*(x1 + y1),(x2 + y2),(x3 + y3),(x4 + y4)*] ) holds
b1 = b2
proof end;
commutativity
for b1 being set
for x, y being quaternion number st ex x1, x2, x3, x4, y1, y2, y3, y4 being Element of REAL st
( x = [*x1,x2,x3,x4*] & y = [*y1,y2,y3,y4*] & b1 = [*(x1 + y1),(x2 + y2),(x3 + y3),(x4 + y4)*] ) holds
ex x1, x2, x3, x4, y1, y2, y3, y4 being Element of REAL st
( y = [*x1,x2,x3,x4*] & x = [*y1,y2,y3,y4*] & b1 = [*(x1 + y1),(x2 + y2),(x3 + y3),(x4 + y4)*] ) ;
end;

:: deftheorem Def7 defines + QUATERNI:def 7 :
for x, y being quaternion number
for b3 being set holds
( b3 = x + y iff ex x1, x2, x3, x4, y1, y2, y3, y4 being Element of REAL st
( x = [*x1,x2,x3,x4*] & y = [*y1,y2,y3,y4*] & b3 = [*(x1 + y1),(x2 + y2),(x3 + y3),(x4 + y4)*] ) );

Lm7: 0 = [*0,0,0,0*]
proof end;

definition
let z be quaternion number ;
consider v, w, x, y being Element of REAL such that
A1: z = [*v,w,x,y*] by Th7;
func - z -> quaternion number means :Def8: :: QUATERNI:def 8
z + it = 0 ;
existence
ex b1 being quaternion number st z + b1 = 0
proof end;
uniqueness
for b1, b2 being quaternion number st z + b1 = 0 & z + b2 = 0 holds
b1 = b2
proof end;
involutiveness
for b1, b2 being quaternion number st b2 + b1 = 0 holds
b1 + b2 = 0 ;
end;

:: deftheorem Def8 defines - QUATERNI:def 8 :
for z, b2 being quaternion number holds
( b2 = - z iff z + b2 = 0 );

definition
let x, y be quaternion number ;
func x - y -> set equals :: QUATERNI:def 9
x + (- y);
coherence
x + (- y) is set ;
end;

:: deftheorem defines - QUATERNI:def 9 :
for x, y being quaternion number holds x - y = x + (- y);

definition
let x, y be quaternion number ;
consider x1, x2, x3, x4 being Element of REAL such that
A1: x = [*x1,x2,x3,x4*] by Th7;
consider y1, y2, y3, y4 being Element of REAL such that
A2: y = [*y1,y2,y3,y4*] by Th7;
func x * y -> set means :Def10: :: QUATERNI:def 10
 ex x1, x2, x3, x4, y1, y2, y3, y4 being Element of REAL st
( x = [*x1,x2,x3,x4*] & y = [*y1,y2,y3,y4*] & it = [*((((x1 * y1) - (x2 * y2)) - (x3 * y3)) - (x4 * y4)),((((x1 * y2) + (x2 * y1)) + (x3 * y4)) - (x4 * y3)),((((x1 * y3) + (y1 * x3)) + (y2 * x4)) - (y4 * x2)),((((x1 * y4) + (x4 * y1)) + (x2 * y3)) - (x3 * y2))*] );
existence
ex b1 being set ex x1, x2, x3, x4, y1, y2, y3, y4 being Element of REAL st
( x = [*x1,x2,x3,x4*] & y = [*y1,y2,y3,y4*] & b1 = [*((((x1 * y1) - (x2 * y2)) - (x3 * y3)) - (x4 * y4)),((((x1 * y2) + (x2 * y1)) + (x3 * y4)) - (x4 * y3)),((((x1 * y3) + (y1 * x3)) + (y2 * x4)) - (y4 * x2)),((((x1 * y4) + (x4 * y1)) + (x2 * y3)) - (x3 * y2))*] )
proof end;
uniqueness
for b1, b2 being set st ex x1, x2, x3, x4, y1, y2, y3, y4 being Element of REAL st
( x = [*x1,x2,x3,x4*] & y = [*y1,y2,y3,y4*] & b1 = [*((((x1 * y1) - (x2 * y2)) - (x3 * y3)) - (x4 * y4)),((((x1 * y2) + (x2 * y1)) + (x3 * y4)) - (x4 * y3)),((((x1 * y3) + (y1 * x3)) + (y2 * x4)) - (y4 * x2)),((((x1 * y4) + (x4 * y1)) + (x2 * y3)) - (x3 * y2))*] ) & ex x1, x2, x3, x4, y1, y2, y3, y4 being Element of REAL st
( x = [*x1,x2,x3,x4*] & y = [*y1,y2,y3,y4*] & b2 = [*((((x1 * y1) - (x2 * y2)) - (x3 * y3)) - (x4 * y4)),((((x1 * y2) + (x2 * y1)) + (x3 * y4)) - (x4 * y3)),((((x1 * y3) + (y1 * x3)) + (y2 * x4)) - (y4 * x2)),((((x1 * y4) + (x4 * y1)) + (x2 * y3)) - (x3 * y2))*] ) holds
b1 = b2
proof end;
end;

:: deftheorem Def10 defines * QUATERNI:def 10 :
for x, y being quaternion number
for b3 being set holds
( b3 = x * y iff ex x1, x2, x3, x4, y1, y2, y3, y4 being Element of REAL st
( x = [*x1,x2,x3,x4*] & y = [*y1,y2,y3,y4*] & b3 = [*((((x1 * y1) - (x2 * y2)) - (x3 * y3)) - (x4 * y4)),((((x1 * y2) + (x2 * y1)) + (x3 * y4)) - (x4 * y3)),((((x1 * y3) + (y1 * x3)) + (y2 * x4)) - (y4 * x2)),((((x1 * y4) + (x4 * y1)) + (x2 * y3)) - (x3 * y2))*] ) );

registration
let z, z9 be quaternion number ;
cluster z + z9 -> quaternion ;
coherence
z + z9 is quaternion
proof end;
cluster z * z9 -> quaternion ;
coherence
z * z9 is quaternion
proof end;
end;

definition
:: original: <j>
redefine func <j> -> Element of QUATERNION equals :: QUATERNI:def 11
[*0,0,1,0*];
coherence
<j> is Element of QUATERNION by Def2;
compatibility
for b1 being Element of QUATERNION holds
( b1 = <j> iff b1 = [*0,0,1,0*] ) by Def6;
:: original: <k>
redefine func <k> -> Element of QUATERNION equals :: QUATERNI:def 12
[*0,0,0,1*];
coherence
<k> is Element of QUATERNION by Def2;
compatibility
for b1 being Element of QUATERNION holds
( b1 = <k> iff b1 = [*0,0,0,1*] ) by Def6;
end;

:: deftheorem defines <j> QUATERNI:def 11 :
<j> = [*0,0,1,0*];

:: deftheorem defines <k> QUATERNI:def 12 :
<k> = [*0,0,0,1*];

theorem :: QUATERNI:13
<i> * <i> = - 1
proof end;

theorem :: QUATERNI:14
<j> * <j> = - 1
proof end;

theorem :: QUATERNI:15
<k> * <k> = - 1
proof end;

theorem :: QUATERNI:16
<i> * <j> = <k>
proof end;

theorem :: QUATERNI:17
<j> * <k> = <i>
proof end;

theorem :: QUATERNI:18
<k> * <i> = <j>
proof end;

theorem :: QUATERNI:19
<i> * <j> = - (<j> * <i>)
proof end;

theorem :: QUATERNI:20
<j> * <k> = - (<k> * <j>)
proof end;

theorem :: QUATERNI:21
<k> * <i> = - (<i> * <k>)
proof end;

definition
let z be quaternion number ;
func Rea z -> set means :Def13: :: QUATERNI:def 13
 ex z9 being complex number st
( z = z9 & it = Re z9 ) if z in COMPLEX
otherwise ex f being Function of 4,REAL st
( z = f & it = f . 0 );
existence
( ( z in COMPLEX implies ex b1 being set ex z9 being complex number st
( z = z9 & b1 = Re z9 ) ) & ( not z in COMPLEX implies ex b1 being set ex f being Function of 4,REAL st
( z = f & b1 = f . 0 ) ) )
proof end;
uniqueness
for b1, b2 being set holds
( ( z in COMPLEX & ex z9 being complex number st
( z = z9 & b1 = Re z9 ) & ex z9 being complex number st
( z = z9 & b2 = Re z9 ) implies b1 = b2 ) & ( not z in COMPLEX & ex f being Function of 4,REAL st
( z = f & b1 = f . 0 ) & ex f being Function of 4,REAL st
( z = f & b2 = f . 0 ) implies b1 = b2 ) ) ;
consistency
for b1 being set holds verum ;
func Im1 z -> set means :Def14: :: QUATERNI:def 14
 ex z9 being complex number st
( z = z9 & it = Im z9 ) if z in COMPLEX
otherwise ex f being Function of 4,REAL st
( z = f & it = f . 1 );
existence
( ( z in COMPLEX implies ex b1 being set ex z9 being complex number st
( z = z9 & b1 = Im z9 ) ) & ( not z in COMPLEX implies ex b1 being set ex f being Function of 4,REAL st
( z = f & b1 = f . 1 ) ) )
proof end;
uniqueness
for b1, b2 being set holds
( ( z in COMPLEX & ex z9 being complex number st
( z = z9 & b1 = Im z9 ) & ex z9 being complex number st
( z = z9 & b2 = Im z9 ) implies b1 = b2 ) & ( not z in COMPLEX & ex f being Function of 4,REAL st
( z = f & b1 = f . 1 ) & ex f being Function of 4,REAL st
( z = f & b2 = f . 1 ) implies b1 = b2 ) ) ;
consistency
for b1 being set holds verum ;
func Im2 z -> set means :Def15: :: QUATERNI:def 15
it = 0 if z in COMPLEX
otherwise ex f being Function of 4,REAL st
( z = f & it = f . 2 );
existence
( ( z in COMPLEX implies ex b1 being set st b1 = 0 ) & ( not z in COMPLEX implies ex b1 being set ex f being Function of 4,REAL st
( z = f & b1 = f . 2 ) ) )
proof end;
uniqueness
for b1, b2 being set holds
( ( z in COMPLEX & b1 = 0 & b2 = 0 implies b1 = b2 ) & ( not z in COMPLEX & ex f being Function of 4,REAL st
( z = f & b1 = f . 2 ) & ex f being Function of 4,REAL st
( z = f & b2 = f . 2 ) implies b1 = b2 ) ) ;
consistency
for b1 being set holds verum ;
func Im3 z -> set means :Def16: :: QUATERNI:def 16
it = 0 if z in COMPLEX
otherwise ex f being Function of 4,REAL st
( z = f & it = f . 3 );
existence
( ( z in COMPLEX implies ex b1 being set st b1 = 0 ) & ( not z in COMPLEX implies ex b1 being set ex f being Function of 4,REAL st
( z = f & b1 = f . 3 ) ) )
proof end;
uniqueness
for b1, b2 being set holds
( ( z in COMPLEX & b1 = 0 & b2 = 0 implies b1 = b2 ) & ( not z in COMPLEX & ex f being Function of 4,REAL st
( z = f & b1 = f . 3 ) & ex f being Function of 4,REAL st
( z = f & b2 = f . 3 ) implies b1 = b2 ) ) ;
consistency
for b1 being set holds verum ;
end;

:: deftheorem Def13 defines Rea QUATERNI:def 13 :
for z being quaternion number
for b2 being set holds
( ( z in COMPLEX implies ( b2 = Rea z iff ex z9 being complex number st
( z = z9 & b2 = Re z9 ) ) ) & ( not z in COMPLEX implies ( b2 = Rea z iff ex f being Function of 4,REAL st
( z = f & b2 = f . 0 ) ) ) );

:: deftheorem Def14 defines Im1 QUATERNI:def 14 :
for z being quaternion number
for b2 being set holds
( ( z in COMPLEX implies ( b2 = Im1 z iff ex z9 being complex number st
( z = z9 & b2 = Im z9 ) ) ) & ( not z in COMPLEX implies ( b2 = Im1 z iff ex f being Function of 4,REAL st
( z = f & b2 = f . 1 ) ) ) );

:: deftheorem Def15 defines Im2 QUATERNI:def 15 :
for z being quaternion number
for b2 being set holds
( ( z in COMPLEX implies ( b2 = Im2 z iff b2 = 0 ) ) & ( not z in COMPLEX implies ( b2 = Im2 z iff ex f being Function of 4,REAL st
( z = f & b2 = f . 2 ) ) ) );

:: deftheorem Def16 defines Im3 QUATERNI:def 16 :
for z being quaternion number
for b2 being set holds
( ( z in COMPLEX implies ( b2 = Im3 z iff b2 = 0 ) ) & ( not z in COMPLEX implies ( b2 = Im3 z iff ex f being Function of 4,REAL st
( z = f & b2 = f . 3 ) ) ) );

registration
let z be quaternion number ;
cluster Rea z -> real ;
coherence
Rea z is real
proof end;
cluster Im1 z -> real ;
coherence
Im1 z is real
proof end;
cluster Im2 z -> real ;
coherence
Im2 z is real
proof end;
cluster Im3 z -> real ;
coherence
Im3 z is real
proof end;
end;

definition
let z be quaternion number ;
:: original: Rea
redefine func Rea z -> Real;
coherence
Rea z is Real by XREAL_0:def 1;
:: original: Im1
redefine func Im1 z -> Real;
coherence
Im1 z is Real by XREAL_0:def 1;
:: original: Im2
redefine func Im2 z -> Real;
coherence
Im2 z is Real by XREAL_0:def 1;
:: original: Im3
redefine func Im3 z -> Real;
coherence
Im3 z is Real by XREAL_0:def 1;
end;

theorem Th22: :: QUATERNI:22
for f being Function of 4,REAL ex a, b, c, d being Element of REAL st f = (0,1,2,3) --> (a,b,c,d)
proof end;

Lm8: for a, b being Element of REAL holds
( Re [*a,b*] = a & Im [*a,b*] = b )
proof end;

Lm9: for z being complex number holds [*(Re z),(Im z)*] = z
proof end;

theorem Th23: :: QUATERNI:23
for a, b, c, d being Element of REAL holds
( Rea [*a,b,c,d*] = a & Im1 [*a,b,c,d*] = b & Im2 [*a,b,c,d*] = c & Im3 [*a,b,c,d*] = d )
proof end;

theorem Th24: :: QUATERNI:24
for z being quaternion number holds z = [*(Rea z),(Im1 z),(Im2 z),(Im3 z)*]
proof end;

theorem Th25: :: QUATERNI:25
for z1, z2 being quaternion number st Rea z1 = Rea z2 & Im1 z1 = Im1 z2 & Im2 z1 = Im2 z2 & Im3 z1 = Im3 z2 holds
z1 = z2
proof end;

definition
func 0q -> quaternion number equals :: QUATERNI:def 17
 0 ;
coherence
0 is quaternion number by Lm7;
func 1q -> quaternion number equals :: QUATERNI:def 18
1;
coherence
1 is quaternion number
proof end;
end;

:: deftheorem defines 0q QUATERNI:def 17 :
0q = 0 ;

:: deftheorem defines 1q QUATERNI:def 18 :
1q = 1;

Lm10: for a, b, c, d being real number st (((a ^2) + (b ^2)) + (c ^2)) + (d ^2) = 0 holds
( a = 0 & b = 0 & c = 0 & d = 0 )
proof end;

theorem Th26: :: QUATERNI:26
for z being quaternion number st Rea z = 0 & Im1 z = 0 & Im2 z = 0 & Im3 z = 0 holds
z = 0q
proof end;

theorem :: QUATERNI:27
for z being quaternion number st z = 0 holds
((((Rea z) ^2) + ((Im1 z) ^2)) + ((Im2 z) ^2)) + ((Im3 z) ^2) = 0
proof end;

theorem :: QUATERNI:28
for z being quaternion number st ((((Rea z) ^2) + ((Im1 z) ^2)) + ((Im2 z) ^2)) + ((Im3 z) ^2) = 0 holds
z = 0q
proof end;

Lm11: [*1,0,0,0*] = 1
proof end;

theorem :: QUATERNI:29
( Rea 1q = 1 & Im1 1q = 0 & Im2 1q = 0 & Im3 1q = 0 ) by Lm11, Th23;

theorem Th30: :: QUATERNI:30
( Rea <i> = 0 & Im1 <i> = 1 & Im2 <i> = 0 & Im3 <i> = 0 )
proof end;

theorem Th31: :: QUATERNI:31
( Rea <j> = 0 & Im1 <j> = 0 & Im2 <j> = 1 & Im3 <j> = 0 & Rea <k> = 0 & Im1 <k> = 0 & Im2 <k> = 0 & Im3 <k> = 1 ) by Th23;

Lm12: for m, n, x, y, z being quaternion number st z = ((m + n) + x) + y holds
( Rea z = (((Rea m) + (Rea n)) + (Rea x)) + (Rea y) & Im1 z = (((Im1 m) + (Im1 n)) + (Im1 x)) + (Im1 y) & Im2 z = (((Im2 m) + (Im2 n)) + (Im2 x)) + (Im2 y) & Im3 z = (((Im3 m) + (Im3 n)) + (Im3 x)) + (Im3 y) )
proof end;

Lm13: for x, y, z being quaternion number st z = x + y holds
( Rea z = (Rea x) + (Rea y) & Im1 z = (Im1 x) + (Im1 y) & Im2 z = (Im2 x) + (Im2 y) & Im3 z = (Im3 x) + (Im3 y) )
proof end;

Lm14: for z1, z2 being quaternion number holds z1 + z2 = [*((Rea z1) + (Rea z2)),((Im1 z1) + (Im1 z2)),((Im2 z1) + (Im2 z2)),((Im3 z1) + (Im3 z2))*]
proof end;

Lm15: for x, y, z being quaternion number st z = x * y holds
( Rea z = ((((Rea x) * (Rea y)) - ((Im1 x) * (Im1 y))) - ((Im2 x) * (Im2 y))) - ((Im3 x) * (Im3 y)) & Im1 z = ((((Rea x) * (Im1 y)) + ((Im1 x) * (Rea y))) + ((Im2 x) * (Im3 y))) - ((Im3 x) * (Im2 y)) & Im2 z = ((((Rea x) * (Im2 y)) + ((Im2 x) * (Rea y))) + ((Im3 x) * (Im1 y))) - ((Im1 x) * (Im3 y)) & Im3 z = ((((Rea x) * (Im3 y)) + ((Im3 x) * (Rea y))) + ((Im1 x) * (Im2 y))) - ((Im2 x) * (Im1 y)) )
proof end;

Lm16: for z1, z2, z3, z4 being quaternion number holds ((z1 + z2) + z3) + z4 = [*((((Rea z1) + (Rea z2)) + (Rea z3)) + (Rea z4)),((((Im1 z1) + (Im1 z2)) + (Im1 z3)) + (Im1 z4)),((((Im2 z1) + (Im2 z2)) + (Im2 z3)) + (Im2 z4)),((((Im3 z1) + (Im3 z2)) + (Im3 z3)) + (Im3 z4))*]
proof end;

Lm17: for z1, z2 being quaternion number holds z1 * z2 = [*(((((Rea z1) * (Rea z2)) - ((Im1 z1) * (Im1 z2))) - ((Im2 z1) * (Im2 z2))) - ((Im3 z1) * (Im3 z2))),(((((Rea z1) * (Im1 z2)) + ((Im1 z1) * (Rea z2))) + ((Im2 z1) * (Im3 z2))) - ((Im3 z1) * (Im2 z2))),(((((Rea z1) * (Im2 z2)) + ((Im2 z1) * (Rea z2))) + ((Im3 z1) * (Im1 z2))) - ((Im1 z1) * (Im3 z2))),(((((Rea z1) * (Im3 z2)) + ((Im3 z1) * (Rea z2))) + ((Im1 z1) * (Im2 z2))) - ((Im2 z1) * (Im1 z2)))*]
proof end;

Lm18: for z1, z2 being quaternion number holds
( Rea (z1 * z2) = ((((Rea z1) * (Rea z2)) - ((Im1 z1) * (Im1 z2))) - ((Im2 z1) * (Im2 z2))) - ((Im3 z1) * (Im3 z2)) & Im1 (z1 * z2) = ((((Rea z1) * (Im1 z2)) + ((Im1 z1) * (Rea z2))) + ((Im2 z1) * (Im3 z2))) - ((Im3 z1) * (Im2 z2)) & Im2 (z1 * z2) = ((((Rea z1) * (Im2 z2)) + ((Im2 z1) * (Rea z2))) + ((Im3 z1) * (Im1 z2))) - ((Im1 z1) * (Im3 z2)) & Im3 (z1 * z2) = ((((Rea z1) * (Im3 z2)) + ((Im3 z1) * (Rea z2))) + ((Im1 z1) * (Im2 z2))) - ((Im2 z1) * (Im1 z2)) )
proof end;

theorem :: QUATERNI:32
for z1, z2, z3, z4 being quaternion number holds
( Rea (((z1 + z2) + z3) + z4) = (((Rea z1) + (Rea z2)) + (Rea z3)) + (Rea z4) & Im1 (((z1 + z2) + z3) + z4) = (((Im1 z1) + (Im1 z2)) + (Im1 z3)) + (Im1 z4) & Im2 (((z1 + z2) + z3) + z4) = (((Im2 z1) + (Im2 z2)) + (Im2 z3)) + (Im2 z4) & Im3 (((z1 + z2) + z3) + z4) = (((Im3 z1) + (Im3 z2)) + (Im3 z3)) + (Im3 z4) )
proof end;

Lm19: for z being quaternion number
for x being Real st z = x holds
( Rea z = x & Im1 z = 0 & Im2 z = 0 & Im3 z = 0 )
proof end;

theorem :: QUATERNI:33
for z1 being quaternion number
for x being Real st z1 = x holds
( Rea (z1 * <i>) = 0 & Im1 (z1 * <i>) = x & Im2 (z1 * <i>) = 0 & Im3 (z1 * <i>) = 0 )
proof end;

theorem :: QUATERNI:34
for z1 being quaternion number
for x being Real st z1 = x holds
( Rea (z1 * <j>) = 0 & Im1 (z1 * <j>) = 0 & Im2 (z1 * <j>) = x & Im3 (z1 * <j>) = 0 )
proof end;

theorem :: QUATERNI:35
for z1 being quaternion number
for x being Real st z1 = x holds
( Rea (z1 * <k>) = 0 & Im1 (z1 * <k>) = 0 & Im2 (z1 * <k>) = 0 & Im3 (z1 * <k>) = x )
proof end;

definition
let x be Real;
let y be quaternion number ;
consider y1, y2, y3, y4 being Element of REAL such that
A1: y = [*y1,y2,y3,y4*] by Th7;
func x + y -> set means :Def19: :: QUATERNI:def 19
 ex y1, y2, y3, y4 being Element of REAL st
( y = [*y1,y2,y3,y4*] & it = [*(x + y1),y2,y3,y4*] );
existence
ex b1 being set ex y1, y2, y3, y4 being Element of REAL st
( y = [*y1,y2,y3,y4*] & b1 = [*(x + y1),y2,y3,y4*] )
proof end;
uniqueness
for b1, b2 being set st ex y1, y2, y3, y4 being Element of REAL st
( y = [*y1,y2,y3,y4*] & b1 = [*(x + y1),y2,y3,y4*] ) & ex y1, y2, y3, y4 being Element of REAL st
( y = [*y1,y2,y3,y4*] & b2 = [*(x + y1),y2,y3,y4*] ) holds
b1 = b2
proof end;
end;

:: deftheorem Def19 defines + QUATERNI:def 19 :
for x being Real
for y being quaternion number
for b3 being set holds
( b3 = x + y iff ex y1, y2, y3, y4 being Element of REAL st
( y = [*y1,y2,y3,y4*] & b3 = [*(x + y1),y2,y3,y4*] ) );

definition
let x be Real;
let y be quaternion number ;
func x - y -> set equals :: QUATERNI:def 20
x + (- y);
coherence
x + (- y) is set ;
end;

:: deftheorem defines - QUATERNI:def 20 :
for x being Real
for y being quaternion number holds x - y = x + (- y);

definition
let x be Real;
let y be quaternion number ;
consider y1, y2, y3, y4 being Element of REAL such that
A1: y = [*y1,y2,y3,y4*] by Th7;
func x * y -> set means :Def21: :: QUATERNI:def 21
 ex y1, y2, y3, y4 being Element of REAL st
( y = [*y1,y2,y3,y4*] & it = [*(x * y1),(x * y2),(x * y3),(x * y4)*] );
existence
ex b1 being set ex y1, y2, y3, y4 being Element of REAL st
( y = [*y1,y2,y3,y4*] & b1 = [*(x * y1),(x * y2),(x * y3),(x * y4)*] )
proof end;
uniqueness
for b1, b2 being set st ex y1, y2, y3, y4 being Element of REAL st
( y = [*y1,y2,y3,y4*] & b1 = [*(x * y1),(x * y2),(x * y3),(x * y4)*] ) & ex y1, y2, y3, y4 being Element of REAL st
( y = [*y1,y2,y3,y4*] & b2 = [*(x * y1),(x * y2),(x * y3),(x * y4)*] ) holds
b1 = b2
proof end;
end;

:: deftheorem Def21 defines * QUATERNI:def 21 :
for x being Real
for y being quaternion number
for b3 being set holds
( b3 = x * y iff ex y1, y2, y3, y4 being Element of REAL st
( y = [*y1,y2,y3,y4*] & b3 = [*(x * y1),(x * y2),(x * y3),(x * y4)*] ) );

registration
let x be Real;
let z9 be quaternion number ;
cluster x + z9 -> quaternion ;
coherence
x + z9 is quaternion
proof end;
cluster x * z9 -> quaternion ;
coherence
x * z9 is quaternion
proof end;
cluster x - z9 -> quaternion ;
coherence
x - z9 is quaternion ;
end;

Lm20: for x, y, z, w being Real holds [*x,y,z,w*] = ((x + (y * <i>)) + (z * <j>)) + (w * <k>)
proof end;

definition
let z1, z2 be quaternion number ;
:: original: +
redefine func z1 + z2 -> Element of QUATERNION ;
coherence
z1 + z2 is Element of QUATERNION by Def2;
end;

Lm21: for z1, z2 being quaternion number holds z1 + z2 = ((((Rea z1) + (Rea z2)) + (((Im1 z1) + (Im1 z2)) * <i>)) + (((Im2 z1) + (Im2 z2)) * <j>)) + (((Im3 z1) + (Im3 z2)) * <k>)
proof end;

theorem Th36: :: QUATERNI:36
for z1, z2 being quaternion number holds
( Rea (z1 + z2) = (Rea z1) + (Rea z2) & Im1 (z1 + z2) = (Im1 z1) + (Im1 z2) & Im2 (z1 + z2) = (Im2 z1) + (Im2 z2) & Im3 (z1 + z2) = (Im3 z1) + (Im3 z2) )
proof end;

definition
let z1, z2 be quaternion number ;
:: original: *
redefine func z1 * z2 -> Element of QUATERNION ;
coherence
z1 * z2 is Element of QUATERNION by Def2;
end;

Lm22: for z1, z2 being quaternion number holds z1 * z2 = (((((((Rea z1) * (Rea z2)) - ((Im1 z1) * (Im1 z2))) - ((Im2 z1) * (Im2 z2))) - ((Im3 z1) * (Im3 z2))) + ((((((Rea z1) * (Im1 z2)) + ((Im1 z1) * (Rea z2))) + ((Im2 z1) * (Im3 z2))) - ((Im3 z1) * (Im2 z2))) * <i>)) + ((((((Rea z1) * (Im2 z2)) + ((Im2 z1) * (Rea z2))) + ((Im3 z1) * (Im1 z2))) - ((Im1 z1) * (Im3 z2))) * <j>)) + ((((((Rea z1) * (Im3 z2)) + ((Im3 z1) * (Rea z2))) + ((Im1 z1) * (Im2 z2))) - ((Im2 z1) * (Im1 z2))) * <k>)
proof end;

theorem :: QUATERNI:37
for z being quaternion number holds z = (((Rea z) + ((Im1 z) * <i>)) + ((Im2 z) * <j>)) + ((Im3 z) * <k>)
proof end;

Lm23: for c being quaternion number holds c + 0q = c
proof end;

Lm24: ( 0 * <i> = 0 & 0 * <j> = 0 & 0 * <k> = 0 )
proof end;

theorem :: QUATERNI:38
for z1, z2 being quaternion number st Im1 z1 = 0 & Im1 z2 = 0 & Im2 z1 = 0 & Im2 z2 = 0 & Im3 z1 = 0 & Im3 z2 = 0 holds
( Rea (z1 * z2) = (Rea z1) * (Rea z2) & Im1 (z1 * z2) = ((Im2 z1) * (Im3 z2)) - ((Im3 z1) * (Im2 z2)) & Im2 (z1 * z2) = ((Im3 z1) * (Im1 z2)) - ((Im1 z1) * (Im3 z2)) & Im3 (z1 * z2) = ((Im1 z1) * (Im2 z2)) - ((Im2 z1) * (Im1 z2)) )
proof end;

theorem :: QUATERNI:39
for z1, z2 being quaternion number st Rea z1 = 0 & Rea z2 = 0 holds
( Rea (z1 * z2) = ((- ((Im1 z1) * (Im1 z2))) - ((Im2 z1) * (Im2 z2))) - ((Im3 z1) * (Im3 z2)) & Im1 (z1 * z2) = ((Im2 z1) * (Im3 z2)) - ((Im3 z1) * (Im2 z2)) & Im2 (z1 * z2) = ((Im3 z1) * (Im1 z2)) - ((Im1 z1) * (Im3 z2)) & Im3 (z1 * z2) = ((Im1 z1) * (Im2 z2)) - ((Im2 z1) * (Im1 z2)) )
proof end;

theorem :: QUATERNI:40
for z being quaternion number holds
( Rea (z * z) = ((((Rea z) ^2) - ((Im1 z) ^2)) - ((Im2 z) ^2)) - ((Im3 z) ^2) & Im1 (z * z) = 2 * ((Rea z) * (Im1 z)) & Im2 (z * z) = 2 * ((Rea z) * (Im2 z)) & Im3 (z * z) = 2 * ((Rea z) * (Im3 z)) )
proof end;

definition
let z be quaternion number ;
:: original: -
redefine func - z -> Element of QUATERNION ;
coherence
- z is Element of QUATERNION by Def2;
end;

Lm25: for z being quaternion number holds - z = (((- (Rea z)) + ((- (Im1 z)) * <i>)) + ((- (Im2 z)) * <j>)) + ((- (Im3 z)) * <k>)
proof end;

theorem Th41: :: QUATERNI:41
for z being quaternion number holds
( Rea (- z) = - (Rea z) & Im1 (- z) = - (Im1 z) & Im2 (- z) = - (Im2 z) & Im3 (- z) = - (Im3 z) )
proof end;

definition
let z1, z2 be quaternion number ;
:: original: -
redefine func z1 - z2 -> Element of QUATERNION ;
coherence
z1 - z2 is Element of QUATERNION by Def2;
end;

Lm26: for z1, z2 being quaternion number holds z1 - z2 = ((((Rea z1) - (Rea z2)) + (((Im1 z1) - (Im1 z2)) * <i>)) + (((Im2 z1) - (Im2 z2)) * <j>)) + (((Im3 z1) - (Im3 z2)) * <k>)
proof end;

theorem Th42: :: QUATERNI:42
for z1, z2 being quaternion number holds
( Rea (z1 - z2) = (Rea z1) - (Rea z2) & Im1 (z1 - z2) = (Im1 z1) - (Im1 z2) & Im2 (z1 - z2) = (Im2 z1) - (Im2 z2) & Im3 (z1 - z2) = (Im3 z1) - (Im3 z2) )
proof end;

definition
let z be quaternion number ;
func z *' -> quaternion number equals :: QUATERNI:def 22
(((Rea z) + ((- (Im1 z)) * <i>)) + ((- (Im2 z)) * <j>)) + ((- (Im3 z)) * <k>);
coherence
(((Rea z) + ((- (Im1 z)) * <i>)) + ((- (Im2 z)) * <j>)) + ((- (Im3 z)) * <k>) is quaternion number ;
end;

:: deftheorem defines *' QUATERNI:def 22 :
for z being quaternion number holds z *' = (((Rea z) + ((- (Im1 z)) * <i>)) + ((- (Im2 z)) * <j>)) + ((- (Im3 z)) * <k>);

definition
let z be quaternion number ;
:: original: *'
redefine func z *' -> Element of QUATERNION ;
coherence
z *' is Element of QUATERNION ;
end;

theorem Th43: :: QUATERNI:43
for z being quaternion number holds z *' = [*(Rea z),(- (Im1 z)),(- (Im2 z)),(- (Im3 z))*]
proof end;

theorem :: QUATERNI:44
for z being quaternion number holds
( Rea (z *') = Rea z & Im1 (z *') = - (Im1 z) & Im2 (z *') = - (Im2 z) & Im3 (z *') = - (Im3 z) )
proof end;

theorem :: QUATERNI:45
for z being quaternion number st z = 0 holds
z *' = 0
proof end;

theorem :: QUATERNI:46
for z being quaternion number st z *' = 0 holds
z = 0
proof end;

theorem :: QUATERNI:47
1q *' = 1q
proof end;

theorem :: QUATERNI:48
( Rea (<i> *') = 0 & Im1 (<i> *') = - 1 & Im2 (<i> *') = 0 & Im3 (<i> *') = 0 )
proof end;

theorem :: QUATERNI:49
( Rea (<j> *') = 0 & Im1 (<j> *') = 0 & Im2 (<j> *') = - 1 & Im3 (<j> *') = 0 )
proof end;

theorem :: QUATERNI:50
( Rea (<k> *') = 0 & Im1 (<k> *') = 0 & Im2 (<k> *') = 0 & Im3 (<k> *') = - 1 )
proof end;

theorem :: QUATERNI:51
<i> *' = - <i> by Th30, Lm25;

theorem :: QUATERNI:52
<j> *' = - <j> by Th31, Lm25;

theorem :: QUATERNI:53
<k> *' = - <k> by Th31, Lm25;

theorem :: QUATERNI:54
for z1, z2 being quaternion number holds (z1 + z2) *' = (z1 *') + (z2 *')
proof end;

theorem Th55: :: QUATERNI:55
for z being quaternion number holds (- z) *' = - (z *')
proof end;

theorem :: QUATERNI:56
for z1, z2 being quaternion number holds (z1 - z2) *' = (z1 *') - (z2 *')
proof end;

theorem :: QUATERNI:57
for z1, z2 being quaternion number st (z1 * z2) *' = (z1 *') * (z2 *') holds
(Im2 z1) * (Im3 z2) = (Im3 z1) * (Im2 z2)
proof end;

theorem :: QUATERNI:58
for z being quaternion number st Im1 z = 0 & Im2 z = 0 & Im3 z = 0 holds
z *' = z
proof end;

theorem :: QUATERNI:59
for z being quaternion number st Rea z = 0 holds
z *' = - z
proof end;

theorem :: QUATERNI:60
for z being quaternion number holds
( Rea (z * (z *')) = ((((Rea z) ^2) + ((Im1 z) ^2)) + ((Im2 z) ^2)) + ((Im3 z) ^2) & Im1 (z * (z *')) = 0 & Im2 (z * (z *')) = 0 & Im3 (z * (z *')) = 0 )
proof end;

theorem :: QUATERNI:61
for z being quaternion number holds
( Rea (z + (z *')) = 2 * (Rea z) & Im1 (z + (z *')) = 0 & Im2 (z + (z *')) = 0 & Im3 (z + (z *')) = 0 )
proof end;

theorem Th62: :: QUATERNI:62
for z being quaternion number holds - z = [*(- (Rea z)),(- (Im1 z)),(- (Im2 z)),(- (Im3 z))*]
proof end;

theorem :: QUATERNI:63
for z1, z2 being quaternion number holds z1 - z2 = [*((Rea z1) - (Rea z2)),((Im1 z1) - (Im1 z2)),((Im2 z1) - (Im2 z2)),((Im3 z1) - (Im3 z2))*]
proof end;

theorem :: QUATERNI:64
for z being quaternion number holds
( Rea (z - (z *')) = 0 & Im1 (z - (z *')) = 2 * (Im1 z) & Im2 (z - (z *')) = 2 * (Im2 z) & Im3 (z - (z *')) = 2 * (Im3 z) )
proof end;

definition
let z be quaternion number ;
func |.z.| -> real number equals :: QUATERNI:def 23
 sqrt (((((Rea z) ^2) + ((Im1 z) ^2)) + ((Im2 z) ^2)) + ((Im3 z) ^2));
coherence
sqrt (((((Rea z) ^2) + ((Im1 z) ^2)) + ((Im2 z) ^2)) + ((Im3 z) ^2)) is real number ;
end;

:: deftheorem defines |. QUATERNI:def 23 :
for z being quaternion number holds |.z.| = sqrt (((((Rea z) ^2) + ((Im1 z) ^2)) + ((Im2 z) ^2)) + ((Im3 z) ^2));

theorem Th65: :: QUATERNI:65
|.0q.| = 0
proof end;

theorem Th66: :: QUATERNI:66
for z being quaternion number st |.z.| = 0 holds
z = 0
proof end;

theorem Th67: :: QUATERNI:67
for z being quaternion number holds 0 <= |.z.|
proof end;

theorem :: QUATERNI:68
|.1q.| = 1
proof end;

theorem :: QUATERNI:69
|.<i>.| = 1 by Th30, SQUARE_1:18;

theorem :: QUATERNI:70
|.<j>.| = 1 by Th31, SQUARE_1:18;

theorem :: QUATERNI:71
|.<k>.| = 1 by Th31, SQUARE_1:18;

theorem Th72: :: QUATERNI:72
for z being quaternion number holds |.(- z).| = |.z.|
proof end;

theorem Th73: :: QUATERNI:73
for z being quaternion number holds |.(z *').| = |.z.|
proof end;

theorem Th74: :: QUATERNI:74
for a, b, c, d being real number holds (((a ^2) + (b ^2)) + (c ^2)) + (d ^2) >= 0
proof end;

Lm27: for a, b, c, d being Element of REAL holds a ^2 <= (((a ^2) + (b ^2)) + (c ^2)) + (d ^2)
proof end;

Lm28: for a, b, c, d being real number holds c ^2 <= (((a ^2) + (b ^2)) + (c ^2)) + (d ^2)
proof end;

Lm29: for d, a, b, c being Element of REAL holds d ^2 <= (((a ^2) + (b ^2)) + (c ^2)) + (d ^2)
proof end;

Lm30: for a, b being Element of REAL st a >= b ^2 holds
sqrt a >= b
proof end;

Lm31: for a1, a2, a3, a4, a5, a6, b1, b2, b3, b4, b5, b6 being real number st a1 >= b1 & a2 >= b2 & a3 >= b3 & a4 >= b4 & a5 >= b5 & a6 >= b6 holds
((((a1 + a2) + a3) + a4) + a5) + a6 >= ((((b1 + b2) + b3) + b4) + b5) + b6
proof end;

Lm32: for a, b being Element of REAL st a >= 0 & b >= 0 & a ^2 >= b ^2 holds
a >= b
proof end;

Lm33: for m1, m2, m4, m3, n1, n2, n3, n4 being real number holds ((sqrt ((((m1 ^2) + (m2 ^2)) + (m3 ^2)) + (m4 ^2))) + (sqrt ((((n1 ^2) + (n2 ^2)) + (n3 ^2)) + (n4 ^2)))) ^2 = ((((((((m1 ^2) + (m2 ^2)) + (m3 ^2)) + (m4 ^2)) + (n1 ^2)) + (n2 ^2)) + (n3 ^2)) + (n4 ^2)) + (2 * (sqrt (((((m1 ^2) + (m2 ^2)) + (m3 ^2)) + (m4 ^2)) * ((((n1 ^2) + (n2 ^2)) + (n3 ^2)) + (n4 ^2)))))
proof end;

Lm34: for m1, m2, m3, m4, n1, n2, n3, n4 being real number holds (sqrt (((((m1 + n1) ^2) + ((m2 + n2) ^2)) + ((m3 + n3) ^2)) + ((m4 + n4) ^2))) ^2 = ((((m1 + n1) ^2) + ((m2 + n2) ^2)) + ((m3 + n3) ^2)) + ((m4 + n4) ^2)
proof end;

theorem :: QUATERNI:75
for z being quaternion number holds Rea z <= |.z.|
proof end;

theorem :: QUATERNI:76
for z being quaternion number holds Im1 z <= |.z.|
proof end;

theorem :: QUATERNI:77
for z being quaternion number holds Im2 z <= |.z.|
proof end;

theorem :: QUATERNI:78
for z being quaternion number holds Im3 z <= |.z.|
proof end;

theorem Th79: :: QUATERNI:79
for z1, z2 being quaternion number holds |.(z1 + z2).| <= |.z1.| + |.z2.|
proof end;

theorem Th80: :: QUATERNI:80
for z1, z2 being quaternion number holds |.(z1 - z2).| <= |.z1.| + |.z2.|
proof end;

Lm35: for z1, z2 being quaternion number holds z1 = (z1 + z2) - z2
proof end;

Lm36: for z1, z2 being quaternion number holds z1 = (z1 - z2) + z2
proof end;

Lm37: for z1, z2 being quaternion number holds z1 - z2 = [*((Rea z1) - (Rea z2)),((Im1 z1) - (Im1 z2)),((Im2 z1) - (Im2 z2)),((Im3 z1) - (Im3 z2))*]
proof end;

Lm38: for z1, z2 being quaternion number holds z1 - z2 = - (z2 - z1)
proof end;

Lm39: for z1, z2 being quaternion number st z1 - z2 = 0 holds
z1 = z2
proof end;

theorem :: QUATERNI:81
for z1, z2 being quaternion number holds |.z1.| - |.z2.| <= |.(z1 + z2).|
proof end;

theorem Th82: :: QUATERNI:82
for z1, z2 being quaternion number holds |.z1.| - |.z2.| <= |.(z1 - z2).|
proof end;

theorem Th83: :: QUATERNI:83
for z1, z2 being quaternion number holds |.(z1 - z2).| = |.(z2 - z1).|
proof end;

theorem :: QUATERNI:84
for z1, z2 being quaternion number holds
( |.(z1 - z2).| = 0 iff z1 = z2 )
proof end;

Lm40: for z, z1, z2 being quaternion number holds z1 - z2 = (z1 - z) + (z - z2)
proof end;

theorem :: QUATERNI:85
for z1, z2, z being quaternion number holds |.(z1 - z2).| <= |.(z1 - z).| + |.(z - z2).|
proof end;

theorem :: QUATERNI:86
for z1, z2 being quaternion number holds |.(|.z1.| - |.z2.|).| <= |.(z1 - z2).|
proof end;

theorem Th87: :: QUATERNI:87
for z1, z2 being quaternion number holds |.(z1 * z2).| = |.z1.| * |.z2.|
proof end;

theorem :: QUATERNI:88
for z being quaternion number holds |.(z * z).| = ((((Rea z) ^2) + ((Im1 z) ^2)) + ((Im2 z) ^2)) + ((Im3 z) ^2)
proof end;

theorem :: QUATERNI:89
for z being quaternion number holds |.(z * z).| = |.(z * (z *')).|
proof end;

theorem :: QUATERNI:90
for z being quaternion number st z is real holds
( Rea z = z & Im1 z = 0 & Im2 z = 0 & Im3 z = 0 )
proof end;

theorem :: QUATERNI:91
for x, y being Element of REAL holds [*x,y,0,0*] = [*x,y*] by Lm4;

theorem :: QUATERNI:92
for z1, z2 being quaternion number holds z1 + z2 = ((((Rea z1) + (Rea z2)) + (((Im1 z1) + (Im1 z2)) * <i>)) + (((Im2 z1) + (Im2 z2)) * <j>)) + (((Im3 z1) + (Im3 z2)) * <k>) by Lm21;

theorem :: QUATERNI:93
for z1, z2 being quaternion number holds z1 * z2 = (((((((Rea z1) * (Rea z2)) - ((Im1 z1) * (Im1 z2))) - ((Im2 z1) * (Im2 z2))) - ((Im3 z1) * (Im3 z2))) + ((((((Rea z1) * (Im1 z2)) + ((Im1 z1) * (Rea z2))) + ((Im2 z1) * (Im3 z2))) - ((Im3 z1) * (Im2 z2))) * <i>)) + ((((((Rea z1) * (Im2 z2)) + ((Im2 z1) * (Rea z2))) + ((Im3 z1) * (Im1 z2))) - ((Im1 z1) * (Im3 z2))) * <j>)) + ((((((Rea z1) * (Im3 z2)) + ((Im3 z1) * (Rea z2))) + ((Im1 z1) * (Im2 z2))) - ((Im2 z1) * (Im1 z2))) * <k>) by Lm22;

theorem :: QUATERNI:94
for z being quaternion number holds - z = (((- (Rea z)) + ((- (Im1 z)) * <i>)) + ((- (Im2 z)) * <j>)) + ((- (Im3 z)) * <k>) by Lm25;

theorem :: QUATERNI:95
for z1, z2 being quaternion number holds z1 - z2 = ((((Rea z1) - (Rea z2)) + (((Im1 z1) - (Im1 z2)) * <i>)) + (((Im2 z1) - (Im2 z2)) * <j>)) + (((Im3 z1) - (Im3 z2)) * <k>) by Lm26;

theorem :: QUATERNI:96
for a, b, c, d being real number st (((a ^2) + (b ^2)) + (c ^2)) + (d ^2) = 0 holds
( a = 0 & b = 0 & c = 0 & d = 0 ) by Lm10;

theorem :: QUATERNI:97
for z1, z2 being quaternion number holds
( Rea (z1 * z2) = ((((Rea z1) * (Rea z2)) - ((Im1 z1) * (Im1 z2))) - ((Im2 z1) * (Im2 z2))) - ((Im3 z1) * (Im3 z2)) & Im1 (z1 * z2) = ((((Rea z1) * (Im1 z2)) + ((Im1 z1) * (Rea z2))) + ((Im2 z1) * (Im3 z2))) - ((Im3 z1) * (Im2 z2)) & Im2 (z1 * z2) = ((((Rea z1) * (Im2 z2)) + ((Im2 z1) * (Rea z2))) + ((Im3 z1) * (Im1 z2))) - ((Im1 z1) * (Im3 z2)) & Im3 (z1 * z2) = ((((Rea z1) * (Im3 z2)) + ((Im3 z1) * (Rea z2))) + ((Im1 z1) * (Im2 z2))) - ((Im2 z1) * (Im1 z2)) ) by Lm18;