set L = LKer f;

set vq = VectQuot (V,(LKer f));

set Cv = CosetSet (V,(LKer f));

set aCv = addCoset (V,(LKer f));

set mCv = lmultCoset (V,(LKer f));

set R = RKer f;

set wq = VectQuot (W,(RKer f));

set Cw = CosetSet (W,(RKer f));

set aCw = addCoset (W,(RKer f));

set mCw = lmultCoset (W,(RKer f));

defpred S_{1}[ set , set , set ] means for v being Vector of V

for w being Vector of W st $1 = v + (LKer f) & $2 = w + (RKer f) holds

$3 = f . (v,w);

A10: for A being Vector of (VectQuot (V,(LKer f)))

for B being Vector of (VectQuot (W,(RKer f))) holds S_{1}[A,B,ff . (A,B)]
from BINOP_1:sch 3(A1);

reconsider ff = ff as Form of (VectQuot (V,(LKer f))),(VectQuot (W,(RKer f))) ;

A11: CosetSet (V,(LKer f)) = the carrier of (VectQuot (V,(LKer f))) by VECTSP10:def 6;

take ff ; :: thesis: for A being Vector of (VectQuot (V,(LKer f)))

for B being Vector of (VectQuot (W,(RKer f)))

for v being Vector of V

for w being Vector of W st A = v + (LKer f) & B = w + (RKer f) holds

ff . (A,B) = f . (v,w)

thus for A being Vector of (VectQuot (V,(LKer f)))

for B being Vector of (VectQuot (W,(RKer f)))

for v being Vector of V

for w being Vector of W st A = v + (LKer f) & B = w + (RKer f) holds

ff . (A,B) = f . (v,w) by A10; :: thesis: verum

set vq = VectQuot (V,(LKer f));

set Cv = CosetSet (V,(LKer f));

set aCv = addCoset (V,(LKer f));

set mCv = lmultCoset (V,(LKer f));

set R = RKer f;

set wq = VectQuot (W,(RKer f));

set Cw = CosetSet (W,(RKer f));

set aCw = addCoset (W,(RKer f));

set mCw = lmultCoset (W,(RKer f));

defpred S

for w being Vector of W st $1 = v + (LKer f) & $2 = w + (RKer f) holds

$3 = f . (v,w);

A1: now :: thesis: for A being Vector of (VectQuot (V,(LKer f)))

for B being Vector of (VectQuot (W,(RKer f))) ex y being Element of the carrier of K st S_{1}[A,B,y]

consider ff being Function of [: the carrier of (VectQuot (V,(LKer f))), the carrier of (VectQuot (W,(RKer f))):], the carrier of K such that for B being Vector of (VectQuot (W,(RKer f))) ex y being Element of the carrier of K st S

let A be Vector of (VectQuot (V,(LKer f))); :: thesis: for B being Vector of (VectQuot (W,(RKer f))) ex y being Element of the carrier of K st S_{1}[A,B,y]

let B be Vector of (VectQuot (W,(RKer f))); :: thesis: ex y being Element of the carrier of K st S_{1}[A,B,y]

consider a being Vector of V such that

A2: A = a + (LKer f) by VECTSP10:22;

consider b being Vector of W such that

A3: B = b + (RKer f) by VECTSP10:22;

take y = f . (a,b); :: thesis: S_{1}[A,B,y]

_{1}[A,B,y]
; :: thesis: verum

end;let B be Vector of (VectQuot (W,(RKer f))); :: thesis: ex y being Element of the carrier of K st S

consider a being Vector of V such that

A2: A = a + (LKer f) by VECTSP10:22;

consider b being Vector of W such that

A3: B = b + (RKer f) by VECTSP10:22;

take y = f . (a,b); :: thesis: S

now :: thesis: for a1 being Vector of V

for b1 being Vector of W st A = a1 + (LKer f) & B = b1 + (RKer f) holds

y = f . (a1,b1)

hence
Sfor b1 being Vector of W st A = a1 + (LKer f) & B = b1 + (RKer f) holds

y = f . (a1,b1)

let a1 be Vector of V; :: thesis: for b1 being Vector of W st A = a1 + (LKer f) & B = b1 + (RKer f) holds

y = f . (a1,b1)

let b1 be Vector of W; :: thesis: ( A = a1 + (LKer f) & B = b1 + (RKer f) implies y = f . (a1,b1) )

assume A = a1 + (LKer f) ; :: thesis: ( B = b1 + (RKer f) implies y = f . (a1,b1) )

then a in a1 + (LKer f) by A2, VECTSP_4:44;

then consider vv being Vector of V such that

A4: vv in LKer f and

A5: a = a1 + vv by VECTSP_4:42;

vv in the carrier of (LKer f) by A4;

then vv in leftker f by Def18;

then A6: ex aa being Vector of V st

( aa = vv & ( for w0 being Vector of W holds f . (aa,w0) = 0. K ) ) ;

assume B = b1 + (RKer f) ; :: thesis: y = f . (a1,b1)

then b in b1 + (RKer f) by A3, VECTSP_4:44;

then consider ww being Vector of W such that

A7: ww in RKer f and

A8: b = b1 + ww by VECTSP_4:42;

ww in the carrier of (RKer f) by A7;

then ww in rightker f by Def19;

then A9: ex bb being Vector of W st

( bb = ww & ( for v0 being Vector of V holds f . (v0,bb) = 0. K ) ) ;

thus y = ((f . (a1,b1)) + (f . (a1,ww))) + ((f . (vv,b1)) + (f . (vv,ww))) by A5, A8, Th28

.= ((f . (a1,b1)) + (0. K)) + ((f . (vv,b1)) + (f . (vv,ww))) by A9

.= ((f . (a1,b1)) + (0. K)) + ((0. K) + (f . (vv,ww))) by A6

.= (f . (a1,b1)) + ((0. K) + (f . (vv,ww))) by RLVECT_1:def 4

.= (f . (a1,b1)) + (f . (vv,ww)) by RLVECT_1:4

.= (f . (a1,b1)) + (0. K) by A6

.= f . (a1,b1) by RLVECT_1:def 4 ; :: thesis: verum

end;y = f . (a1,b1)

let b1 be Vector of W; :: thesis: ( A = a1 + (LKer f) & B = b1 + (RKer f) implies y = f . (a1,b1) )

assume A = a1 + (LKer f) ; :: thesis: ( B = b1 + (RKer f) implies y = f . (a1,b1) )

then a in a1 + (LKer f) by A2, VECTSP_4:44;

then consider vv being Vector of V such that

A4: vv in LKer f and

A5: a = a1 + vv by VECTSP_4:42;

vv in the carrier of (LKer f) by A4;

then vv in leftker f by Def18;

then A6: ex aa being Vector of V st

( aa = vv & ( for w0 being Vector of W holds f . (aa,w0) = 0. K ) ) ;

assume B = b1 + (RKer f) ; :: thesis: y = f . (a1,b1)

then b in b1 + (RKer f) by A3, VECTSP_4:44;

then consider ww being Vector of W such that

A7: ww in RKer f and

A8: b = b1 + ww by VECTSP_4:42;

ww in the carrier of (RKer f) by A7;

then ww in rightker f by Def19;

then A9: ex bb being Vector of W st

( bb = ww & ( for v0 being Vector of V holds f . (v0,bb) = 0. K ) ) ;

thus y = ((f . (a1,b1)) + (f . (a1,ww))) + ((f . (vv,b1)) + (f . (vv,ww))) by A5, A8, Th28

.= ((f . (a1,b1)) + (0. K)) + ((f . (vv,b1)) + (f . (vv,ww))) by A9

.= ((f . (a1,b1)) + (0. K)) + ((0. K) + (f . (vv,ww))) by A6

.= (f . (a1,b1)) + ((0. K) + (f . (vv,ww))) by RLVECT_1:def 4

.= (f . (a1,b1)) + (f . (vv,ww)) by RLVECT_1:4

.= (f . (a1,b1)) + (0. K) by A6

.= f . (a1,b1) by RLVECT_1:def 4 ; :: thesis: verum

A10: for A being Vector of (VectQuot (V,(LKer f)))

for B being Vector of (VectQuot (W,(RKer f))) holds S

reconsider ff = ff as Form of (VectQuot (V,(LKer f))),(VectQuot (W,(RKer f))) ;

A11: CosetSet (V,(LKer f)) = the carrier of (VectQuot (V,(LKer f))) by VECTSP10:def 6;

A12: now :: thesis: for ww being Vector of (VectQuot (W,(RKer f))) holds FunctionalSAF (ff,ww) is homogeneous

A16:
CosetSet (W,(RKer f)) = the carrier of (VectQuot (W,(RKer f)))
by VECTSP10:def 6;let ww be Vector of (VectQuot (W,(RKer f))); :: thesis: FunctionalSAF (ff,ww) is homogeneous

consider w being Vector of W such that

A13: ww = w + (RKer f) by VECTSP10:22;

set ffw = FunctionalSAF (ff,ww);

end;consider w being Vector of W such that

A13: ww = w + (RKer f) by VECTSP10:22;

set ffw = FunctionalSAF (ff,ww);

now :: thesis: for A being Vector of (VectQuot (V,(LKer f)))

for r being Element of K holds (FunctionalSAF (ff,ww)) . (r * A) = r * ((FunctionalSAF (ff,ww)) . A)

hence
FunctionalSAF (ff,ww) is homogeneous
; :: thesis: verumfor r being Element of K holds (FunctionalSAF (ff,ww)) . (r * A) = r * ((FunctionalSAF (ff,ww)) . A)

let A be Vector of (VectQuot (V,(LKer f))); :: thesis: for r being Element of K holds (FunctionalSAF (ff,ww)) . (r * A) = r * ((FunctionalSAF (ff,ww)) . A)

let r be Element of K; :: thesis: (FunctionalSAF (ff,ww)) . (r * A) = r * ((FunctionalSAF (ff,ww)) . A)

consider a being Vector of V such that

A14: A = a + (LKer f) by VECTSP10:22;

A15: ( the lmult of (VectQuot (V,(LKer f))) = lmultCoset (V,(LKer f)) & (lmultCoset (V,(LKer f))) . (r,A) = (r * a) + (LKer f) ) by A11, A14, VECTSP10:def 5, VECTSP10:def 6;

thus (FunctionalSAF (ff,ww)) . (r * A) = ff . ((r * A),ww) by Th9

.= f . ((r * a),w) by A10, A13, A15

.= r * (f . (a,w)) by Th31

.= r * (ff . (A,ww)) by A10, A13, A14

.= r * ((FunctionalSAF (ff,ww)) . A) by Th9 ; :: thesis: verum

end;let r be Element of K; :: thesis: (FunctionalSAF (ff,ww)) . (r * A) = r * ((FunctionalSAF (ff,ww)) . A)

consider a being Vector of V such that

A14: A = a + (LKer f) by VECTSP10:22;

A15: ( the lmult of (VectQuot (V,(LKer f))) = lmultCoset (V,(LKer f)) & (lmultCoset (V,(LKer f))) . (r,A) = (r * a) + (LKer f) ) by A11, A14, VECTSP10:def 5, VECTSP10:def 6;

thus (FunctionalSAF (ff,ww)) . (r * A) = ff . ((r * A),ww) by Th9

.= f . ((r * a),w) by A10, A13, A15

.= r * (f . (a,w)) by Th31

.= r * (ff . (A,ww)) by A10, A13, A14

.= r * ((FunctionalSAF (ff,ww)) . A) by Th9 ; :: thesis: verum

A17: now :: thesis: for vv being Vector of (VectQuot (V,(LKer f))) holds FunctionalFAF (ff,vv) is homogeneous

let vv be Vector of (VectQuot (V,(LKer f))); :: thesis: FunctionalFAF (ff,vv) is homogeneous

consider v being Vector of V such that

A18: vv = v + (LKer f) by VECTSP10:22;

set ffv = FunctionalFAF (ff,vv);

end;consider v being Vector of V such that

A18: vv = v + (LKer f) by VECTSP10:22;

set ffv = FunctionalFAF (ff,vv);

now :: thesis: for A being Vector of (VectQuot (W,(RKer f)))

for r being Element of K holds (FunctionalFAF (ff,vv)) . (r * A) = r * ((FunctionalFAF (ff,vv)) . A)

hence
FunctionalFAF (ff,vv) is homogeneous
; :: thesis: verumfor r being Element of K holds (FunctionalFAF (ff,vv)) . (r * A) = r * ((FunctionalFAF (ff,vv)) . A)

let A be Vector of (VectQuot (W,(RKer f))); :: thesis: for r being Element of K holds (FunctionalFAF (ff,vv)) . (r * A) = r * ((FunctionalFAF (ff,vv)) . A)

let r be Element of K; :: thesis: (FunctionalFAF (ff,vv)) . (r * A) = r * ((FunctionalFAF (ff,vv)) . A)

consider a being Vector of W such that

A19: A = a + (RKer f) by VECTSP10:22;

A20: ( the lmult of (VectQuot (W,(RKer f))) = lmultCoset (W,(RKer f)) & (lmultCoset (W,(RKer f))) . (r,A) = (r * a) + (RKer f) ) by A16, A19, VECTSP10:def 5, VECTSP10:def 6;

thus (FunctionalFAF (ff,vv)) . (r * A) = ff . (vv,(r * A)) by Th8

.= f . (v,(r * a)) by A10, A18, A20

.= r * (f . (v,a)) by Th32

.= r * (ff . (vv,A)) by A10, A18, A19

.= r * ((FunctionalFAF (ff,vv)) . A) by Th8 ; :: thesis: verum

end;let r be Element of K; :: thesis: (FunctionalFAF (ff,vv)) . (r * A) = r * ((FunctionalFAF (ff,vv)) . A)

consider a being Vector of W such that

A19: A = a + (RKer f) by VECTSP10:22;

A20: ( the lmult of (VectQuot (W,(RKer f))) = lmultCoset (W,(RKer f)) & (lmultCoset (W,(RKer f))) . (r,A) = (r * a) + (RKer f) ) by A16, A19, VECTSP10:def 5, VECTSP10:def 6;

thus (FunctionalFAF (ff,vv)) . (r * A) = ff . (vv,(r * A)) by Th8

.= f . (v,(r * a)) by A10, A18, A20

.= r * (f . (v,a)) by Th32

.= r * (ff . (vv,A)) by A10, A18, A19

.= r * ((FunctionalFAF (ff,vv)) . A) by Th8 ; :: thesis: verum

A21: now :: thesis: for ww being Vector of (VectQuot (W,(RKer f))) holds FunctionalSAF (ff,ww) is additive

let ww be Vector of (VectQuot (W,(RKer f))); :: thesis: FunctionalSAF (ff,ww) is additive

consider w being Vector of W such that

A22: ww = w + (RKer f) by VECTSP10:22;

set ffw = FunctionalSAF (ff,ww);

end;consider w being Vector of W such that

A22: ww = w + (RKer f) by VECTSP10:22;

set ffw = FunctionalSAF (ff,ww);

now :: thesis: for A, B being Vector of (VectQuot (V,(LKer f))) holds (FunctionalSAF (ff,ww)) . (A + B) = ((FunctionalSAF (ff,ww)) . A) + ((FunctionalSAF (ff,ww)) . B)

hence
FunctionalSAF (ff,ww) is additive
; :: thesis: verumlet A, B be Vector of (VectQuot (V,(LKer f))); :: thesis: (FunctionalSAF (ff,ww)) . (A + B) = ((FunctionalSAF (ff,ww)) . A) + ((FunctionalSAF (ff,ww)) . B)

consider a being Vector of V such that

A23: A = a + (LKer f) by VECTSP10:22;

consider b being Vector of V such that

A24: B = b + (LKer f) by VECTSP10:22;

A25: ( the addF of (VectQuot (V,(LKer f))) = addCoset (V,(LKer f)) & (addCoset (V,(LKer f))) . (A,B) = (a + b) + (LKer f) ) by A11, A23, A24, VECTSP10:def 3, VECTSP10:def 6;

thus (FunctionalSAF (ff,ww)) . (A + B) = ff . ((A + B),ww) by Th9

.= f . ((a + b),w) by A10, A22, A25, RLVECT_1:2

.= (f . (a,w)) + (f . (b,w)) by Th26

.= (ff . (A,ww)) + (f . (b,w)) by A10, A22, A23

.= (ff . (A,ww)) + (ff . (B,ww)) by A10, A22, A24

.= ((FunctionalSAF (ff,ww)) . A) + (ff . (B,ww)) by Th9

.= ((FunctionalSAF (ff,ww)) . A) + ((FunctionalSAF (ff,ww)) . B) by Th9 ; :: thesis: verum

end;consider a being Vector of V such that

A23: A = a + (LKer f) by VECTSP10:22;

consider b being Vector of V such that

A24: B = b + (LKer f) by VECTSP10:22;

A25: ( the addF of (VectQuot (V,(LKer f))) = addCoset (V,(LKer f)) & (addCoset (V,(LKer f))) . (A,B) = (a + b) + (LKer f) ) by A11, A23, A24, VECTSP10:def 3, VECTSP10:def 6;

thus (FunctionalSAF (ff,ww)) . (A + B) = ff . ((A + B),ww) by Th9

.= f . ((a + b),w) by A10, A22, A25, RLVECT_1:2

.= (f . (a,w)) + (f . (b,w)) by Th26

.= (ff . (A,ww)) + (f . (b,w)) by A10, A22, A23

.= (ff . (A,ww)) + (ff . (B,ww)) by A10, A22, A24

.= ((FunctionalSAF (ff,ww)) . A) + (ff . (B,ww)) by Th9

.= ((FunctionalSAF (ff,ww)) . A) + ((FunctionalSAF (ff,ww)) . B) by Th9 ; :: thesis: verum

now :: thesis: for vv being Vector of (VectQuot (V,(LKer f))) holds FunctionalFAF (ff,vv) is additive

then reconsider ff = ff as bilinear-Form of (VectQuot (V,(LKer f))),(VectQuot (W,(RKer f))) by A21, A12, A17, Def11, Def12, Def13, Def14;let vv be Vector of (VectQuot (V,(LKer f))); :: thesis: FunctionalFAF (ff,vv) is additive

consider v being Vector of V such that

A26: vv = v + (LKer f) by VECTSP10:22;

set ffv = FunctionalFAF (ff,vv);

end;consider v being Vector of V such that

A26: vv = v + (LKer f) by VECTSP10:22;

set ffv = FunctionalFAF (ff,vv);

now :: thesis: for A, B being Vector of (VectQuot (W,(RKer f))) holds (FunctionalFAF (ff,vv)) . (A + B) = ((FunctionalFAF (ff,vv)) . A) + ((FunctionalFAF (ff,vv)) . B)

hence
FunctionalFAF (ff,vv) is additive
; :: thesis: verumlet A, B be Vector of (VectQuot (W,(RKer f))); :: thesis: (FunctionalFAF (ff,vv)) . (A + B) = ((FunctionalFAF (ff,vv)) . A) + ((FunctionalFAF (ff,vv)) . B)

consider a being Vector of W such that

A27: A = a + (RKer f) by VECTSP10:22;

consider b being Vector of W such that

A28: B = b + (RKer f) by VECTSP10:22;

A29: ( the addF of (VectQuot (W,(RKer f))) = addCoset (W,(RKer f)) & (addCoset (W,(RKer f))) . (A,B) = (a + b) + (RKer f) ) by A16, A27, A28, VECTSP10:def 3, VECTSP10:def 6;

thus (FunctionalFAF (ff,vv)) . (A + B) = ff . (vv,(A + B)) by Th8

.= f . (v,(a + b)) by A10, A26, A29, RLVECT_1:2

.= (f . (v,a)) + (f . (v,b)) by Th27

.= (ff . (vv,A)) + (f . (v,b)) by A10, A26, A27

.= (ff . (vv,A)) + (ff . (vv,B)) by A10, A26, A28

.= ((FunctionalFAF (ff,vv)) . A) + (ff . (vv,B)) by Th8

.= ((FunctionalFAF (ff,vv)) . A) + ((FunctionalFAF (ff,vv)) . B) by Th8 ; :: thesis: verum

end;consider a being Vector of W such that

A27: A = a + (RKer f) by VECTSP10:22;

consider b being Vector of W such that

A28: B = b + (RKer f) by VECTSP10:22;

A29: ( the addF of (VectQuot (W,(RKer f))) = addCoset (W,(RKer f)) & (addCoset (W,(RKer f))) . (A,B) = (a + b) + (RKer f) ) by A16, A27, A28, VECTSP10:def 3, VECTSP10:def 6;

thus (FunctionalFAF (ff,vv)) . (A + B) = ff . (vv,(A + B)) by Th8

.= f . (v,(a + b)) by A10, A26, A29, RLVECT_1:2

.= (f . (v,a)) + (f . (v,b)) by Th27

.= (ff . (vv,A)) + (f . (v,b)) by A10, A26, A27

.= (ff . (vv,A)) + (ff . (vv,B)) by A10, A26, A28

.= ((FunctionalFAF (ff,vv)) . A) + (ff . (vv,B)) by Th8

.= ((FunctionalFAF (ff,vv)) . A) + ((FunctionalFAF (ff,vv)) . B) by Th8 ; :: thesis: verum

take ff ; :: thesis: for A being Vector of (VectQuot (V,(LKer f)))

for B being Vector of (VectQuot (W,(RKer f)))

for v being Vector of V

for w being Vector of W st A = v + (LKer f) & B = w + (RKer f) holds

ff . (A,B) = f . (v,w)

thus for A being Vector of (VectQuot (V,(LKer f)))

for B being Vector of (VectQuot (W,(RKer f)))

for v being Vector of V

for w being Vector of W st A = v + (LKer f) & B = w + (RKer f) holds

ff . (A,B) = f . (v,w) by A10; :: thesis: verum