let V be non empty Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct ; :: thesis: for V1 being non empty Subset of
for d1 being Element of V1
for A being BinOp of V1
for M being Function of [:REAL ,V1:],V1 st d1 = 0. V & A = the addF of V || V1 & M = the Mult of V | [:REAL ,V1:] holds
( RLSStruct(# V1,d1,A,M #) is Abelian & RLSStruct(# V1,d1,A,M #) is add-associative & RLSStruct(# V1,d1,A,M #) is right_zeroed & RLSStruct(# V1,d1,A,M #) is RealLinearSpace-like )
let V1 be non empty Subset of ; :: thesis: for d1 being Element of V1
for A being BinOp of V1
for M being Function of [:REAL ,V1:],V1 st d1 = 0. V & A = the addF of V || V1 & M = the Mult of V | [:REAL ,V1:] holds
( RLSStruct(# V1,d1,A,M #) is Abelian & RLSStruct(# V1,d1,A,M #) is add-associative & RLSStruct(# V1,d1,A,M #) is right_zeroed & RLSStruct(# V1,d1,A,M #) is RealLinearSpace-like )
let d1 be Element of V1; :: thesis: for A being BinOp of V1
for M being Function of [:REAL ,V1:],V1 st d1 = 0. V & A = the addF of V || V1 & M = the Mult of V | [:REAL ,V1:] holds
( RLSStruct(# V1,d1,A,M #) is Abelian & RLSStruct(# V1,d1,A,M #) is add-associative & RLSStruct(# V1,d1,A,M #) is right_zeroed & RLSStruct(# V1,d1,A,M #) is RealLinearSpace-like )
let A be BinOp of V1; :: thesis: for M being Function of [:REAL ,V1:],V1 st d1 = 0. V & A = the addF of V || V1 & M = the Mult of V | [:REAL ,V1:] holds
( RLSStruct(# V1,d1,A,M #) is Abelian & RLSStruct(# V1,d1,A,M #) is add-associative & RLSStruct(# V1,d1,A,M #) is right_zeroed & RLSStruct(# V1,d1,A,M #) is RealLinearSpace-like )
let M be Function of [:REAL ,V1:],V1; :: thesis: ( d1 = 0. V & A = the addF of V || V1 & M = the Mult of V | [:REAL ,V1:] implies ( RLSStruct(# V1,d1,A,M #) is Abelian & RLSStruct(# V1,d1,A,M #) is add-associative & RLSStruct(# V1,d1,A,M #) is right_zeroed & RLSStruct(# V1,d1,A,M #) is RealLinearSpace-like ) )
set D = V1;
assume that
A1: d1 = 0. V and
A2: A = the addF of V || V1 and
A3: M = the Mult of V | [:REAL ,V1:] ; :: thesis: ( RLSStruct(# V1,d1,A,M #) is Abelian & RLSStruct(# V1,d1,A,M #) is add-associative & RLSStruct(# V1,d1,A,M #) is right_zeroed & RLSStruct(# V1,d1,A,M #) is RealLinearSpace-like )
set W = RLSStruct(# V1,d1,A,M #);

A4: now

let a be Real; :: thesis: for x being VECTOR of holds a * x = the Mult of V . a,x
let x be VECTOR of ; :: thesis: a * x = the Mult of V . a,x
thus a * x = the Mult of V . [a,x] by A3, FUNCT_1:72
.= the Mult of V . a,x ; :: thesis: verum

end;

A5: now

let x, y be VECTOR of ; :: thesis: x + y = the addF of V . x,y
thus x + y = the addF of V . [x,y] by A2, FUNCT_1:72
.= the addF of V . x,y ; :: thesis: verum

end;

( RLSStruct(# V1,d1,A,M #) is Abelian & RLSStruct(# V1,d1,A,M #) is add-associative & RLSStruct(# V1,d1,A,M #) is right_zeroed & RLSStruct(# V1,d1,A,M #) is RealLinearSpace-like )

proof

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

hereby :: according to RLVECT_1:def 5 :: thesis: ( RLSStruct(# V1,d1,A,M #) is add-associative & RLSStruct(# V1,d1,A,M #) is right_zeroed & RLSStruct(# V1,d1,A,M #) is RealLinearSpace-like )

let x, y be VECTOR of ; :: thesis: x + y = y + x
reconsider x1 = x, y1 = y as VECTOR of by TARSKI:def 3;
thus x + y = x1 + y1 by A5
.= y1 + x1
.= y + x by A5 ; :: thesis: verum

end;

now

let x, y, z be VECTOR of ; :: thesis: (x + y) + z = x + (y + z)
reconsider x1 = x, y1 = y, z1 = z as VECTOR of by TARSKI:def 3;
thus (x + y) + z = the addF of V . (x + y),z1 by A5
.= (x1 + y1) + z1 by A5
.= x1 + (y1 + z1) by RLVECT_1:def 6
.= the addF of V . x1,(y + z) by A5
.= x + (y + z) by A5 ; :: thesis: verum

end;

hence RLSStruct(# V1,d1,A,M #) is add-associative by RLVECT_1:def 6; :: thesis: ( RLSStruct(# V1,d1,A,M #) is right_zeroed & RLSStruct(# V1,d1,A,M #) is RealLinearSpace-like )

now

let x be VECTOR of ; :: thesis: x + (0. RLSStruct(# V1,d1,A,M #)) = x
reconsider y = x as VECTOR of by TARSKI:def 3;
thus x + (0. RLSStruct(# V1,d1,A,M #)) = y + (0. V) by A1, A5
.= x by RLVECT_1:def 7 ; :: thesis: verum

end;

hence RLSStruct(# V1,d1,A,M #) is right_zeroed by RLVECT_1:def 7; :: thesis: RLSStruct(# V1,d1,A,M #) is RealLinearSpace-like

hereby :: according to RLVECT_1:def 9 :: thesis: ( ( for b₁, b₂ being set
for b₃ being Element of the carrier of RLSStruct(# V1,d1,A,M #) holds (b₁ + b₂) * b₃ = (b₁ * b₃) + (b₂ * b₃) ) & ( for b₁, b₂ being set
for b₃ being Element of the carrier of RLSStruct(# V1,d1,A,M #) holds (b₁ * b₂) * b₃ = b₁ * (b₂ * b₃) ) & ( for b₁ being Element of the carrier of RLSStruct(# V1,d1,A,M #) holds 1 * b₁ = b₁ ) )

let a1 be real number ; :: thesis: for x, y being VECTOR of holds a1 * (x + y) = (a1 * x) + (a1 * y)
let x, y be VECTOR of ; :: thesis: a1 * (x + y) = (a1 * x) + (a1 * y)
reconsider a = a1 as Real by XREAL_0:def 1;
reconsider x1 = x, y1 = y as VECTOR of by TARSKI:def 3;
a * (x + y) = the Mult of V . a,(x + y) by A4
.= a * (x1 + y1) by A5
.= (a * x1) + (a * y1) by RLVECT_1:def 9
.= the addF of V . (the Mult of V . a,x1),(a * y) by A4
.= the addF of V . (a * x),(a * y) by A4
.= (a * x) + (a * y) by A5 ;
hence a1 * (x + y) = (a1 * x) + (a1 * y) ; :: thesis: verum

end;

hereby :: thesis: ( ( for b₁, b₂ being set
for b₃ being Element of the carrier of RLSStruct(# V1,d1,A,M #) holds (b₁ * b₂) * b₃ = b₁ * (b₂ * b₃) ) & ( for b₁ being Element of the carrier of RLSStruct(# V1,d1,A,M #) holds 1 * b₁ = b₁ ) )

let a1, b1 be real number ; :: thesis: for x being VECTOR of holds (a1 + b1) * x = (a1 * x) + (b1 * x)
let x be VECTOR of ; :: thesis: (a1 + b1) * x = (a1 * x) + (b1 * x)
reconsider a = a1, b = b1 as Real by XREAL_0:def 1;
reconsider y = x as VECTOR of by TARSKI:def 3;
(a + b) * x = (a + b) * y by A4
.= (a * y) + (b * y) by RLVECT_1:def 9
.= the addF of V . (the Mult of V . a,y),(b * x) by A4
.= the addF of V . (a * x),(b * x) by A4
.= (a * x) + (b * x) by A5 ;
hence (a1 + b1) * x = (a1 * x) + (b1 * x) ; :: thesis: verum

end;

hereby :: thesis: for b₁ being Element of the carrier of RLSStruct(# V1,d1,A,M #) holds 1 * b₁ = b₁

let a1, b1 be real number ; :: thesis: for x being VECTOR of holds (a1 * b1) * x = a1 * (b1 * x)
let x be VECTOR of ; :: thesis: (a1 * b1) * x = a1 * (b1 * x)
reconsider a = a1, b = b1 as Real by XREAL_0:def 1;
reconsider y = x as VECTOR of by TARSKI:def 3;
(a * b) * x = (a * b) * y by A4
.= a * (b * y) by RLVECT_1:def 9
.= the Mult of V . a,(b * x) by A4
.= a * (b * x) by A4 ;
hence (a1 * b1) * x = a1 * (b1 * x) ; :: thesis: verum

end;

let x be VECTOR of ; :: thesis: 1 * x = x
reconsider y = x as VECTOR of by TARSKI:def 3;
thus 1 * x = 1 * y by A4
.= x by RLVECT_1:def 9 ; :: thesis: verum

end;

hence ( RLSStruct(# V1,d1,A,M #) is Abelian & RLSStruct(# V1,d1,A,M #) is add-associative & RLSStruct(# V1,d1,A,M #) is right_zeroed & RLSStruct(# V1,d1,A,M #) is RealLinearSpace-like ) ; :: thesis: verum