let X be non empty set ; :: thesis:
let V be ComplexAlgebra; :: thesis:
let V1 be non empty Subset of V; :: thesis:
let d1, d2 be Element of X; :: thesis:
let A be BinOp of X; :: thesis:
let M be Function of [:X,X:],X; :: thesis:
let MR be Function of [:COMPLEX,X:],X; :: thesis:
assume that
A1: V1 = X and
A2: d1 = 0. V and
A3: d2 = 1. V and
A4: A = the addF of V || V1 and
A5: M = the multF of V || V1 and
A6: MR = the Mult of V | [:COMPLEX,V1:] and
A7: for v being Element of V st v in V1 holds
- v in V1 ; :: according to C0SP1:def 1 :: thesis:
reconsider W = ComplexAlgebraStr(# X,M,A,MR,d2,d1 #) as non empty ComplexAlgebraStr ;

A9: now

let x, y be Element of W; :: thesis:
reconsider x1 = x, y1 = y as Element of V by A1, TARSKI:def 3;
x + y = the addF of V . [x,y] by A1, A4, FUNCT_1:49;
hence x + y = the addF of V . (x,y) ; :: thesis:

end;

A10: now

let a, x be VECTOR of W; :: thesis:
a * x = the multF of V . [a,x] by A1, A5, FUNCT_1:49;
hence a * x = the multF of V . (a,x) ; :: thesis:

end;

A12: ( W is Abelian & W is add-associative & W is right_zeroed & W is right_complementable & W is commutative & W is associative & W is right_unital & W is right-distributive & W is vector-distributive & W is scalar-distributive & W is scalar-associative & W is vector-associative )

proof

set Mv = the multF of V;
set MV = the Mult of V;
set Av = the addF of V;

hereby :: according to RLVECT_1:def 2 :: thesis:

let x, y be VECTOR of W; :: thesis:
reconsider x1 = x, y1 = y as VECTOR of V by A1, TARSKI:def 3;
( x + y = x1 + y1 & y + x = y1 + x1 ) by A9;
hence x + y = y + x ; :: thesis:

end;

hereby :: according to RLVECT_1:def 3 :: thesis:

let x, y, z be VECTOR of W; :: thesis:
reconsider x1 = x, y1 = y, z1 = z as VECTOR of V by A1, TARSKI:def 3;
x + (y + z) = the addF of V . (x1,(y + z)) by A9;
then A13: x + (y + z) = x1 + (y1 + z1) by A9;
(x + y) + z = the addF of V . ((x + y),z1) by A9;
then (x + y) + z = (x1 + y1) + z1 by A9;
hence (x + y) + z = x + (y + z) by A13, RLVECT_1:def 3; :: thesis:

end;

hereby :: according to RLVECT_1:def 4 :: thesis:

let x be VECTOR of W; :: thesis:
reconsider y = x, z = 0. W as VECTOR of V by A1, TARSKI:def 3;
thus x + (0. W) = y + (0. V) by A2, A9
.= x by RLVECT_1:4 ; :: thesis:

end;

thus W is right_complementable :: thesis:

proof

let x be Element of W; :: according to ALGSTR_0:def 16 :: thesis:
reconsider x1 = x as Element of V by A1, TARSKI:def 3;
consider v being Element of V such that
A14: x1 + v = 0. V by ALGSTR_0:def 11;
v = - x1 by A14, VECTSP_1:16;
then reconsider y = v as Element of W by A1, A7;
take y ; :: according to ALGSTR_0:def 11 :: thesis:
thus x + y = 0. W by A2, A9, A14; :: thesis:

end;

hereby :: according to GROUP_1:def 12 :: thesis:

let v, w be Element of W; :: thesis:
reconsider v1 = v, w1 = w as Element of V by A1, TARSKI:def 3;
( v * w = v1 * w1 & w * v = w1 * v1 ) by A10;
hence v * w = w * v ; :: thesis:

end;

hereby :: according to GROUP_1:def 3 :: thesis:

let a, b, x be Element of W; :: thesis:
reconsider y = x, a1 = a, b1 = b as Element of V by A1, TARSKI:def 3;
a * (b * x) = the multF of V . (a,(b * x)) by A10;
then A15: a * (b * x) = a1 * (b1 * y) by A10;
a * b = a1 * b1 by A10;
then (a * b) * x = (a1 * b1) * y by A10;
hence (a * b) * x = a * (b * x) by A15, GROUP_1:def 3; :: thesis:

end;

hereby :: according to VECTSP_1:def 4 :: thesis:

let v be Element of W; :: thesis:
reconsider v1 = v as Element of V by A1, TARSKI:def 3;
v * (1. W) = v1 * (1. V) by A3, A10;
hence v * (1. W) = v by VECTSP_1:def 4; :: thesis:

end;

hereby :: according to VECTSP_1:def 2 :: thesis:

let x, y, z be Element of W; :: thesis:
reconsider x1 = x, y1 = y, z1 = z as Element of V by A1, TARSKI:def 3;
y + z = y1 + z1 by A9;
then x * (y + z) = x1 * (y1 + z1) by A10;
then A16: x * (y + z) = (x1 * y1) + (x1 * z1) by VECTSP_1:def 2;
( x * y = x1 * y1 & x * z = x1 * z1 ) by A10;
hence x * (y + z) = (x * y) + (x * z) by A9, A16; :: thesis:

end;

C: for a being Complex
for x, y being VECTOR of W holds a * (x + y) = (a * x) + (a * y)

proof

let a be Complex; :: thesis:
let x, y be VECTOR of W; :: thesis:
reconsider x1 = x, y1 = y as Element of V by A1, TARSKI:def 3;
A17: a * x = the Mult of V . (a,x) by A1, A6, FUNCT_1:49
.= a * x1 ;
A18: a * y = the Mult of V . (a,y) by A1, A6, FUNCT_1:49
.= a * y1 ;
a * (x + y) = the Mult of V . (a,(x + y)) by A1, A6, FUNCT_1:49
.= a * (x1 + y1) by A9 ;
then a * (x + y) = (a * x1) + (a * y1) by CLVECT_1:def 2;
hence a * (x + y) = (a * x) + (a * y) by A9, A17, A18; :: thesis:

end;

B: for a, b being Complex
for v being VECTOR of W holds (a + b) * v = (a * v) + (b * v)

proof

let a, b be Complex; :: thesis:
let x be Element of W; :: thesis:
reconsider x1 = x as Element of V by A1, TARSKI:def 3;
A21: a * x = the Mult of V . (a,x) by A1, A6, FUNCT_1:49
.= a * x1 ;
A22: b * x = the Mult of V . (b,x) by A1, A6, FUNCT_1:49
.= b * x1 ;
(a + b) * x = the Mult of V . ((a + b),x) by A1, A6, FUNCT_1:49
.= (a + b) * x1 ;
then (a + b) * x = (a * x1) + (b * x1) by CLVECT_1:def 3;
hence (a + b) * x = (a * x) + (b * x) by A9, A21, A22; :: thesis:

end;

A: for a, b being Complex
for v being VECTOR of W holds (a * b) * v = a * (b * v)

proof

let a, b be Complex; :: thesis:
let x be Element of W; :: thesis:
reconsider x1 = x as Element of V by A1, TARSKI:def 3;
reconsider y = b * x as Element of W ;
reconsider y1 = b * x1 as Element of V ;
A24: b * x = the Mult of V . (b,x) by A1, A6, FUNCT_1:49
.= b * x1 ;
A25: a * (b * x) = the Mult of V . (a,y) by A1, A6, FUNCT_1:49
.= a * (b * x1) by A24 ;
(a * b) * x = the Mult of V . ((a * b),x) by A1, A6, FUNCT_1:49
.= (a * b) * x1 ;
hence (a * b) * x = a * (b * x) by A25, CLVECT_1:def 4; :: thesis:

end;

for x, y being Element of W
for a being Complex holds a * (x * y) = (a * x) * y

proof

let x, y be Element of W; :: thesis:
let a be Complex; :: thesis:
reconsider x1 = x, y1 = y as Element of V by A1, TARSKI:def 3;
set z = x * y;
set z1 = x1 * y1;
A28: a * x = the Mult of V . (a,x) by A1, A6, FUNCT_1:49
.= a * x1 ;
A30: a * (x * y) = the Mult of V . (a,(x * y)) by A1, A6, FUNCT_1:49
.= a * (x1 * y1) by A10 ;
a * (x1 * y1) = (a * x1) * y1 by CFUNCDOM:def 9;
hence a * (x * y) = (a * x) * y by A10, A28, A30; :: thesis:

end;

hence ( W is vector-distributive & W is scalar-distributive & W is scalar-associative & W is vector-associative ) by C, B, A, CFUNCDOM:def 9, CLVECT_1:def 2, CLVECT_1:def 3, CLVECT_1:def 4; :: thesis:

end;

( 0. W = 0. V & 1. W = 1. V ) by A2, A3;
hence ComplexAlgebraStr(# X,M,A,MR,d2,d1 #) is ComplexSubAlgebra of V by A1, A4, A5, A6, A12, DefComplexAlgebra; :: thesis: