set A = R_Algebra_of_BoundedLinearOperators X;

set BLOP = BoundedLinearOperators (X,X);

set RRL = RLSStruct(# (BoundedLinearOperators (X,X)),(Zero_ ((BoundedLinearOperators (X,X)),(R_VectorSpace_of_LinearOperators (X,X)))),(Add_ ((BoundedLinearOperators (X,X)),(R_VectorSpace_of_LinearOperators (X,X)))),(Mult_ ((BoundedLinearOperators (X,X)),(R_VectorSpace_of_LinearOperators (X,X)))) #);

set MULT = FuncMult X;

set UNIT = FuncUnit X;

set ADD = Add_ ((BoundedLinearOperators (X,X)),(R_VectorSpace_of_LinearOperators (X,X)));

thus R_Algebra_of_BoundedLinearOperators X is Abelian :: thesis: ( R_Algebra_of_BoundedLinearOperators X is add-associative & R_Algebra_of_BoundedLinearOperators X is right_zeroed & R_Algebra_of_BoundedLinearOperators X is right_complementable & R_Algebra_of_BoundedLinearOperators X is associative & R_Algebra_of_BoundedLinearOperators X is right_unital & R_Algebra_of_BoundedLinearOperators X is right-distributive & R_Algebra_of_BoundedLinearOperators X is vector-distributive & R_Algebra_of_BoundedLinearOperators X is scalar-distributive & R_Algebra_of_BoundedLinearOperators X is scalar-associative & R_Algebra_of_BoundedLinearOperators X is vector-associative & R_Algebra_of_BoundedLinearOperators X is strict )

set BLOP = BoundedLinearOperators (X,X);

set RRL = RLSStruct(# (BoundedLinearOperators (X,X)),(Zero_ ((BoundedLinearOperators (X,X)),(R_VectorSpace_of_LinearOperators (X,X)))),(Add_ ((BoundedLinearOperators (X,X)),(R_VectorSpace_of_LinearOperators (X,X)))),(Mult_ ((BoundedLinearOperators (X,X)),(R_VectorSpace_of_LinearOperators (X,X)))) #);

set MULT = FuncMult X;

set UNIT = FuncUnit X;

set ADD = Add_ ((BoundedLinearOperators (X,X)),(R_VectorSpace_of_LinearOperators (X,X)));

thus R_Algebra_of_BoundedLinearOperators X is Abelian :: thesis: ( R_Algebra_of_BoundedLinearOperators X is add-associative & R_Algebra_of_BoundedLinearOperators X is right_zeroed & R_Algebra_of_BoundedLinearOperators X is right_complementable & R_Algebra_of_BoundedLinearOperators X is associative & R_Algebra_of_BoundedLinearOperators X is right_unital & R_Algebra_of_BoundedLinearOperators X is right-distributive & R_Algebra_of_BoundedLinearOperators X is vector-distributive & R_Algebra_of_BoundedLinearOperators X is scalar-distributive & R_Algebra_of_BoundedLinearOperators X is scalar-associative & R_Algebra_of_BoundedLinearOperators X is vector-associative & R_Algebra_of_BoundedLinearOperators X is strict )

proof

thus
R_Algebra_of_BoundedLinearOperators X is add-associative
:: thesis: ( R_Algebra_of_BoundedLinearOperators X is right_zeroed & R_Algebra_of_BoundedLinearOperators X is right_complementable & R_Algebra_of_BoundedLinearOperators X is associative & R_Algebra_of_BoundedLinearOperators X is right_unital & R_Algebra_of_BoundedLinearOperators X is right-distributive & R_Algebra_of_BoundedLinearOperators X is vector-distributive & R_Algebra_of_BoundedLinearOperators X is scalar-distributive & R_Algebra_of_BoundedLinearOperators X is scalar-associative & R_Algebra_of_BoundedLinearOperators X is vector-associative & R_Algebra_of_BoundedLinearOperators X is strict )
let x, y be Element of (R_Algebra_of_BoundedLinearOperators X); :: according to RLVECT_1:def 2 :: thesis: x + y = y + x

reconsider f = x, g = y as Element of RLSStruct(# (BoundedLinearOperators (X,X)),(Zero_ ((BoundedLinearOperators (X,X)),(R_VectorSpace_of_LinearOperators (X,X)))),(Add_ ((BoundedLinearOperators (X,X)),(R_VectorSpace_of_LinearOperators (X,X)))),(Mult_ ((BoundedLinearOperators (X,X)),(R_VectorSpace_of_LinearOperators (X,X)))) #) ;

thus x + y = f + g

.= y + x by RLVECT_1:2 ; :: thesis: verum

end;reconsider f = x, g = y as Element of RLSStruct(# (BoundedLinearOperators (X,X)),(Zero_ ((BoundedLinearOperators (X,X)),(R_VectorSpace_of_LinearOperators (X,X)))),(Add_ ((BoundedLinearOperators (X,X)),(R_VectorSpace_of_LinearOperators (X,X)))),(Mult_ ((BoundedLinearOperators (X,X)),(R_VectorSpace_of_LinearOperators (X,X)))) #) ;

thus x + y = f + g

.= y + x by RLVECT_1:2 ; :: thesis: verum

proof

thus
R_Algebra_of_BoundedLinearOperators X is right_zeroed
:: thesis: ( R_Algebra_of_BoundedLinearOperators X is right_complementable & R_Algebra_of_BoundedLinearOperators X is associative & R_Algebra_of_BoundedLinearOperators X is right_unital & R_Algebra_of_BoundedLinearOperators X is right-distributive & R_Algebra_of_BoundedLinearOperators X is vector-distributive & R_Algebra_of_BoundedLinearOperators X is scalar-distributive & R_Algebra_of_BoundedLinearOperators X is scalar-associative & R_Algebra_of_BoundedLinearOperators X is vector-associative & R_Algebra_of_BoundedLinearOperators X is strict )
let x, y, z be Element of (R_Algebra_of_BoundedLinearOperators X); :: according to RLVECT_1:def 3 :: thesis: (x + y) + z = x + (y + z)

reconsider f = x, g = y, h = z as Element of RLSStruct(# (BoundedLinearOperators (X,X)),(Zero_ ((BoundedLinearOperators (X,X)),(R_VectorSpace_of_LinearOperators (X,X)))),(Add_ ((BoundedLinearOperators (X,X)),(R_VectorSpace_of_LinearOperators (X,X)))),(Mult_ ((BoundedLinearOperators (X,X)),(R_VectorSpace_of_LinearOperators (X,X)))) #) ;

thus (x + y) + z = (f + g) + h

.= f + (g + h) by RLVECT_1:def 3

.= x + (y + z) ; :: thesis: verum

end;reconsider f = x, g = y, h = z as Element of RLSStruct(# (BoundedLinearOperators (X,X)),(Zero_ ((BoundedLinearOperators (X,X)),(R_VectorSpace_of_LinearOperators (X,X)))),(Add_ ((BoundedLinearOperators (X,X)),(R_VectorSpace_of_LinearOperators (X,X)))),(Mult_ ((BoundedLinearOperators (X,X)),(R_VectorSpace_of_LinearOperators (X,X)))) #) ;

thus (x + y) + z = (f + g) + h

.= f + (g + h) by RLVECT_1:def 3

.= x + (y + z) ; :: thesis: verum

proof

thus
R_Algebra_of_BoundedLinearOperators X is right_complementable
:: thesis: ( R_Algebra_of_BoundedLinearOperators X is associative & R_Algebra_of_BoundedLinearOperators X is right_unital & R_Algebra_of_BoundedLinearOperators X is right-distributive & R_Algebra_of_BoundedLinearOperators X is vector-distributive & R_Algebra_of_BoundedLinearOperators X is scalar-distributive & R_Algebra_of_BoundedLinearOperators X is scalar-associative & R_Algebra_of_BoundedLinearOperators X is vector-associative & R_Algebra_of_BoundedLinearOperators X is strict )
let x be Element of (R_Algebra_of_BoundedLinearOperators X); :: according to RLVECT_1:def 4 :: thesis: x + (0. (R_Algebra_of_BoundedLinearOperators X)) = x

reconsider f = x as Element of RLSStruct(# (BoundedLinearOperators (X,X)),(Zero_ ((BoundedLinearOperators (X,X)),(R_VectorSpace_of_LinearOperators (X,X)))),(Add_ ((BoundedLinearOperators (X,X)),(R_VectorSpace_of_LinearOperators (X,X)))),(Mult_ ((BoundedLinearOperators (X,X)),(R_VectorSpace_of_LinearOperators (X,X)))) #) ;

thus x + (0. (R_Algebra_of_BoundedLinearOperators X)) = f + (0. RLSStruct(# (BoundedLinearOperators (X,X)),(Zero_ ((BoundedLinearOperators (X,X)),(R_VectorSpace_of_LinearOperators (X,X)))),(Add_ ((BoundedLinearOperators (X,X)),(R_VectorSpace_of_LinearOperators (X,X)))),(Mult_ ((BoundedLinearOperators (X,X)),(R_VectorSpace_of_LinearOperators (X,X)))) #))

.= x ; :: thesis: verum

end;reconsider f = x as Element of RLSStruct(# (BoundedLinearOperators (X,X)),(Zero_ ((BoundedLinearOperators (X,X)),(R_VectorSpace_of_LinearOperators (X,X)))),(Add_ ((BoundedLinearOperators (X,X)),(R_VectorSpace_of_LinearOperators (X,X)))),(Mult_ ((BoundedLinearOperators (X,X)),(R_VectorSpace_of_LinearOperators (X,X)))) #) ;

thus x + (0. (R_Algebra_of_BoundedLinearOperators X)) = f + (0. RLSStruct(# (BoundedLinearOperators (X,X)),(Zero_ ((BoundedLinearOperators (X,X)),(R_VectorSpace_of_LinearOperators (X,X)))),(Add_ ((BoundedLinearOperators (X,X)),(R_VectorSpace_of_LinearOperators (X,X)))),(Mult_ ((BoundedLinearOperators (X,X)),(R_VectorSpace_of_LinearOperators (X,X)))) #))

.= x ; :: thesis: verum

proof

thus
R_Algebra_of_BoundedLinearOperators X is associative
:: thesis: ( R_Algebra_of_BoundedLinearOperators X is right_unital & R_Algebra_of_BoundedLinearOperators X is right-distributive & R_Algebra_of_BoundedLinearOperators X is vector-distributive & R_Algebra_of_BoundedLinearOperators X is scalar-distributive & R_Algebra_of_BoundedLinearOperators X is scalar-associative & R_Algebra_of_BoundedLinearOperators X is vector-associative & R_Algebra_of_BoundedLinearOperators X is strict )
let x be Element of (R_Algebra_of_BoundedLinearOperators X); :: according to ALGSTR_0:def 16 :: thesis: x is right_complementable

reconsider f = x as Element of RLSStruct(# (BoundedLinearOperators (X,X)),(Zero_ ((BoundedLinearOperators (X,X)),(R_VectorSpace_of_LinearOperators (X,X)))),(Add_ ((BoundedLinearOperators (X,X)),(R_VectorSpace_of_LinearOperators (X,X)))),(Mult_ ((BoundedLinearOperators (X,X)),(R_VectorSpace_of_LinearOperators (X,X)))) #) ;

consider s being Element of RLSStruct(# (BoundedLinearOperators (X,X)),(Zero_ ((BoundedLinearOperators (X,X)),(R_VectorSpace_of_LinearOperators (X,X)))),(Add_ ((BoundedLinearOperators (X,X)),(R_VectorSpace_of_LinearOperators (X,X)))),(Mult_ ((BoundedLinearOperators (X,X)),(R_VectorSpace_of_LinearOperators (X,X)))) #) such that

A1: f + s = 0. RLSStruct(# (BoundedLinearOperators (X,X)),(Zero_ ((BoundedLinearOperators (X,X)),(R_VectorSpace_of_LinearOperators (X,X)))),(Add_ ((BoundedLinearOperators (X,X)),(R_VectorSpace_of_LinearOperators (X,X)))),(Mult_ ((BoundedLinearOperators (X,X)),(R_VectorSpace_of_LinearOperators (X,X)))) #) by ALGSTR_0:def 11;

reconsider t = s as Element of (R_Algebra_of_BoundedLinearOperators X) ;

take t ; :: according to ALGSTR_0:def 11 :: thesis: x + t = 0. (R_Algebra_of_BoundedLinearOperators X)

thus x + t = 0. (R_Algebra_of_BoundedLinearOperators X) by A1; :: thesis: verum

end;reconsider f = x as Element of RLSStruct(# (BoundedLinearOperators (X,X)),(Zero_ ((BoundedLinearOperators (X,X)),(R_VectorSpace_of_LinearOperators (X,X)))),(Add_ ((BoundedLinearOperators (X,X)),(R_VectorSpace_of_LinearOperators (X,X)))),(Mult_ ((BoundedLinearOperators (X,X)),(R_VectorSpace_of_LinearOperators (X,X)))) #) ;

consider s being Element of RLSStruct(# (BoundedLinearOperators (X,X)),(Zero_ ((BoundedLinearOperators (X,X)),(R_VectorSpace_of_LinearOperators (X,X)))),(Add_ ((BoundedLinearOperators (X,X)),(R_VectorSpace_of_LinearOperators (X,X)))),(Mult_ ((BoundedLinearOperators (X,X)),(R_VectorSpace_of_LinearOperators (X,X)))) #) such that

A1: f + s = 0. RLSStruct(# (BoundedLinearOperators (X,X)),(Zero_ ((BoundedLinearOperators (X,X)),(R_VectorSpace_of_LinearOperators (X,X)))),(Add_ ((BoundedLinearOperators (X,X)),(R_VectorSpace_of_LinearOperators (X,X)))),(Mult_ ((BoundedLinearOperators (X,X)),(R_VectorSpace_of_LinearOperators (X,X)))) #) by ALGSTR_0:def 11;

reconsider t = s as Element of (R_Algebra_of_BoundedLinearOperators X) ;

take t ; :: according to ALGSTR_0:def 11 :: thesis: x + t = 0. (R_Algebra_of_BoundedLinearOperators X)

thus x + t = 0. (R_Algebra_of_BoundedLinearOperators X) by A1; :: thesis: verum

proof

thus
R_Algebra_of_BoundedLinearOperators X is right_unital
:: thesis: ( R_Algebra_of_BoundedLinearOperators X is right-distributive & R_Algebra_of_BoundedLinearOperators X is vector-distributive & R_Algebra_of_BoundedLinearOperators X is scalar-distributive & R_Algebra_of_BoundedLinearOperators X is scalar-associative & R_Algebra_of_BoundedLinearOperators X is vector-associative & R_Algebra_of_BoundedLinearOperators X is strict )
let x, y, z be Element of (R_Algebra_of_BoundedLinearOperators X); :: according to GROUP_1:def 3 :: thesis: (x * y) * z = x * (y * z)

reconsider xx = x, yy = y, zz = z as Element of BoundedLinearOperators (X,X) ;

thus (x * y) * z = (FuncMult X) . ((xx * yy),zz) by Def4

.= (xx * yy) * zz by Def4

.= xx * (yy * zz) by Th7

.= (FuncMult X) . (xx,(yy * zz)) by Def4

.= x * (y * z) by Def4 ; :: thesis: verum

end;reconsider xx = x, yy = y, zz = z as Element of BoundedLinearOperators (X,X) ;

thus (x * y) * z = (FuncMult X) . ((xx * yy),zz) by Def4

.= (xx * yy) * zz by Def4

.= xx * (yy * zz) by Th7

.= (FuncMult X) . (xx,(yy * zz)) by Def4

.= x * (y * z) by Def4 ; :: thesis: verum

proof

thus
R_Algebra_of_BoundedLinearOperators X is right-distributive
:: thesis: ( R_Algebra_of_BoundedLinearOperators X is vector-distributive & R_Algebra_of_BoundedLinearOperators X is scalar-distributive & R_Algebra_of_BoundedLinearOperators X is scalar-associative & R_Algebra_of_BoundedLinearOperators X is vector-associative & R_Algebra_of_BoundedLinearOperators X is strict )
let x be Element of (R_Algebra_of_BoundedLinearOperators X); :: according to VECTSP_1:def 4 :: thesis: x * (1. (R_Algebra_of_BoundedLinearOperators X)) = x

reconsider xx = x as Element of BoundedLinearOperators (X,X) ;

thus x * (1. (R_Algebra_of_BoundedLinearOperators X)) = xx * (FuncUnit X) by Def4

.= x by Th8 ; :: thesis: verum

end;reconsider xx = x as Element of BoundedLinearOperators (X,X) ;

thus x * (1. (R_Algebra_of_BoundedLinearOperators X)) = xx * (FuncUnit X) by Def4

.= x by Th8 ; :: thesis: verum

proof

thus
R_Algebra_of_BoundedLinearOperators X is vector-distributive
:: thesis: ( R_Algebra_of_BoundedLinearOperators X is scalar-distributive & R_Algebra_of_BoundedLinearOperators X is scalar-associative & R_Algebra_of_BoundedLinearOperators X is vector-associative & R_Algebra_of_BoundedLinearOperators X is strict )
let x, y, z be Element of (R_Algebra_of_BoundedLinearOperators X); :: according to VECTSP_1:def 2 :: thesis: x * (y + z) = (x * y) + (x * z)

reconsider xx = x, yy = y, zz = z as Element of BoundedLinearOperators (X,X) ;

thus x * (y + z) = xx * (yy + zz) by Def4

.= (xx * yy) + (xx * zz) by Th9

.= (Add_ ((BoundedLinearOperators (X,X)),(R_VectorSpace_of_LinearOperators (X,X)))) . ((xx * yy),((FuncMult X) . (xx,zz))) by Def4

.= (x * y) + (x * z) by Def4 ; :: thesis: verum

end;reconsider xx = x, yy = y, zz = z as Element of BoundedLinearOperators (X,X) ;

thus x * (y + z) = xx * (yy + zz) by Def4

.= (xx * yy) + (xx * zz) by Th9

.= (Add_ ((BoundedLinearOperators (X,X)),(R_VectorSpace_of_LinearOperators (X,X)))) . ((xx * yy),((FuncMult X) . (xx,zz))) by Def4

.= (x * y) + (x * z) by Def4 ; :: thesis: verum

proof

thus
R_Algebra_of_BoundedLinearOperators X is scalar-distributive
:: thesis: ( R_Algebra_of_BoundedLinearOperators X is scalar-associative & R_Algebra_of_BoundedLinearOperators X is vector-associative & R_Algebra_of_BoundedLinearOperators X is strict )
let a be Real; :: according to RLVECT_1:def 5 :: thesis: for b_{1}, b_{2} being Element of the carrier of (R_Algebra_of_BoundedLinearOperators X) holds a * (b_{1} + b_{2}) = (a * b_{1}) + (a * b_{2})

let x, y be Element of (R_Algebra_of_BoundedLinearOperators X); :: thesis: a * (x + y) = (a * x) + (a * y)

reconsider f = x, g = y as Element of RLSStruct(# (BoundedLinearOperators (X,X)),(Zero_ ((BoundedLinearOperators (X,X)),(R_VectorSpace_of_LinearOperators (X,X)))),(Add_ ((BoundedLinearOperators (X,X)),(R_VectorSpace_of_LinearOperators (X,X)))),(Mult_ ((BoundedLinearOperators (X,X)),(R_VectorSpace_of_LinearOperators (X,X)))) #) ;

thus a * (x + y) = a * (f + g)

.= (a * f) + (a * g) by RLVECT_1:def 5

.= (a * x) + (a * y) ; :: thesis: verum

end;let x, y be Element of (R_Algebra_of_BoundedLinearOperators X); :: thesis: a * (x + y) = (a * x) + (a * y)

reconsider f = x, g = y as Element of RLSStruct(# (BoundedLinearOperators (X,X)),(Zero_ ((BoundedLinearOperators (X,X)),(R_VectorSpace_of_LinearOperators (X,X)))),(Add_ ((BoundedLinearOperators (X,X)),(R_VectorSpace_of_LinearOperators (X,X)))),(Mult_ ((BoundedLinearOperators (X,X)),(R_VectorSpace_of_LinearOperators (X,X)))) #) ;

thus a * (x + y) = a * (f + g)

.= (a * f) + (a * g) by RLVECT_1:def 5

.= (a * x) + (a * y) ; :: thesis: verum

proof

thus
R_Algebra_of_BoundedLinearOperators X is scalar-associative
:: thesis: ( R_Algebra_of_BoundedLinearOperators X is vector-associative & R_Algebra_of_BoundedLinearOperators X is strict )
let a, b be Real; :: according to RLVECT_1:def 6 :: thesis: for b_{1} being Element of the carrier of (R_Algebra_of_BoundedLinearOperators X) holds (a + b) * b_{1} = (a * b_{1}) + (b * b_{1})

let x be Element of (R_Algebra_of_BoundedLinearOperators X); :: thesis: (a + b) * x = (a * x) + (b * x)

reconsider f = x as Element of RLSStruct(# (BoundedLinearOperators (X,X)),(Zero_ ((BoundedLinearOperators (X,X)),(R_VectorSpace_of_LinearOperators (X,X)))),(Add_ ((BoundedLinearOperators (X,X)),(R_VectorSpace_of_LinearOperators (X,X)))),(Mult_ ((BoundedLinearOperators (X,X)),(R_VectorSpace_of_LinearOperators (X,X)))) #) ;

thus (a + b) * x = (a + b) * f

.= (a * f) + (b * f) by RLVECT_1:def 6

.= (a * x) + (b * x) ; :: thesis: verum

end;let x be Element of (R_Algebra_of_BoundedLinearOperators X); :: thesis: (a + b) * x = (a * x) + (b * x)

reconsider f = x as Element of RLSStruct(# (BoundedLinearOperators (X,X)),(Zero_ ((BoundedLinearOperators (X,X)),(R_VectorSpace_of_LinearOperators (X,X)))),(Add_ ((BoundedLinearOperators (X,X)),(R_VectorSpace_of_LinearOperators (X,X)))),(Mult_ ((BoundedLinearOperators (X,X)),(R_VectorSpace_of_LinearOperators (X,X)))) #) ;

thus (a + b) * x = (a + b) * f

.= (a * f) + (b * f) by RLVECT_1:def 6

.= (a * x) + (b * x) ; :: thesis: verum

proof

thus
R_Algebra_of_BoundedLinearOperators X is vector-associative
:: thesis: R_Algebra_of_BoundedLinearOperators X is strict
let a, b be Real; :: according to RLVECT_1:def 7 :: thesis: for b_{1} being Element of the carrier of (R_Algebra_of_BoundedLinearOperators X) holds (a * b) * b_{1} = a * (b * b_{1})

let x be Element of (R_Algebra_of_BoundedLinearOperators X); :: thesis: (a * b) * x = a * (b * x)

reconsider f = x as Element of RLSStruct(# (BoundedLinearOperators (X,X)),(Zero_ ((BoundedLinearOperators (X,X)),(R_VectorSpace_of_LinearOperators (X,X)))),(Add_ ((BoundedLinearOperators (X,X)),(R_VectorSpace_of_LinearOperators (X,X)))),(Mult_ ((BoundedLinearOperators (X,X)),(R_VectorSpace_of_LinearOperators (X,X)))) #) ;

thus (a * b) * x = (a * b) * f

.= a * (b * f) by RLVECT_1:def 7

.= a * (b * x) ; :: thesis: verum

end;let x be Element of (R_Algebra_of_BoundedLinearOperators X); :: thesis: (a * b) * x = a * (b * x)

reconsider f = x as Element of RLSStruct(# (BoundedLinearOperators (X,X)),(Zero_ ((BoundedLinearOperators (X,X)),(R_VectorSpace_of_LinearOperators (X,X)))),(Add_ ((BoundedLinearOperators (X,X)),(R_VectorSpace_of_LinearOperators (X,X)))),(Mult_ ((BoundedLinearOperators (X,X)),(R_VectorSpace_of_LinearOperators (X,X)))) #) ;

thus (a * b) * x = (a * b) * f

.= a * (b * f) by RLVECT_1:def 7

.= a * (b * x) ; :: thesis: verum

proof

thus
R_Algebra_of_BoundedLinearOperators X is strict
; :: thesis: verum
let x, y be Element of (R_Algebra_of_BoundedLinearOperators X); :: according to FUNCSDOM:def 9 :: thesis: for b_{1} being object holds b_{1} * (x * y) = (b_{1} * x) * y

let a be Real; :: thesis: a * (x * y) = (a * x) * y

reconsider xx = x, yy = y as Element of BoundedLinearOperators (X,X) ;

thus a * (x * y) = a * (xx * yy) by Def4

.= (a * xx) * yy by Th12

.= (a * x) * y by Def4 ; :: thesis: verum

end;let a be Real; :: thesis: a * (x * y) = (a * x) * y

reconsider xx = x, yy = y as Element of BoundedLinearOperators (X,X) ;

thus a * (x * y) = a * (xx * yy) by Def4

.= (a * xx) * yy by Th12

.= (a * x) * y by Def4 ; :: thesis: verum