set K = F_Complex ;
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 S1[ 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);
A1: rightker f = rightker (f *') by Th55;
A2: 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 F_Complex st S1[A,B,y]
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 F_Complex st S1[A,B,y]
let B be Vector of (VectQuot (W,(RKer (f *')))); :: thesis: ex y being Element of the carrier of F_Complex st S1[A,B,y]
consider a being Vector of V such that
A3: A = a + (LKer f) by VECTSP10:22;
consider b being Vector of W such that
A4: B = b + (RKer (f *')) by VECTSP10:22;
take y = f . (a,b); :: thesis: S1[A,B,y]
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)
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 A3, VECTSP_4:44;
then consider vv being Vector of V such that
A5: vv in LKer f and
A6: a = a1 + vv by VECTSP_4:42;
vv in the carrier of (LKer f) by A5, STRUCT_0:def 5;
then vv in leftker f by BILINEAR:def 18;
then A7: ex aa being Vector of V st
( aa = vv & ( for w0 being Vector of W holds f . (aa,w0) = 0. F_Complex ) ) ;
assume B = b1 + (RKer (f *')) ; :: thesis: y = f . (a1,b1)
then b in b1 + (RKer (f *')) by A4, VECTSP_4:44;
then consider ww being Vector of W such that
A8: ww in RKer (f *') and
A9: b = b1 + ww by VECTSP_4:42;
ww in the carrier of (RKer (f *')) by A8, STRUCT_0:def 5;
then ww in rightker (f *') by BILINEAR:def 19;
then A10: ex bb being Vector of W st
( bb = ww & ( for v0 being Vector of V holds f . (v0,bb) = 0. F_Complex ) ) by A1;
thus y = ((f . (a1,b1)) + (f . (a1,ww))) + ((f . (vv,b1)) + (f . (vv,ww))) by A6, A9, BILINEAR:28
.= ((f . (a1,b1)) + (0. F_Complex)) + ((f . (vv,b1)) + (f . (vv,ww))) by A10
.= ((f . (a1,b1)) + (0. F_Complex)) + ((0. F_Complex) + (f . (vv,ww))) by A7
.= (f . (a1,b1)) + ((0. F_Complex) + (f . (vv,ww))) by RLVECT_1:def 4
.= (f . (a1,b1)) + (f . (vv,ww)) by RLVECT_1:4
.= (f . (a1,b1)) + (0. F_Complex) by A7
.= f . (a1,b1) by RLVECT_1:def 4 ; :: thesis: verum
end;
hence S1[A,B,y] ; :: thesis: verum
end;
consider ff being Function of [: the carrier of (VectQuot (V,(LKer f))), the carrier of (VectQuot (W,(RKer (f *')))):], the carrier of F_Complex such that
A11: for A being Vector of (VectQuot (V,(LKer f)))
for B being Vector of (VectQuot (W,(RKer (f *')))) holds S1[A,B,ff . (A,B)] from BINOP_1:sch 3(A2);
 reconsider ff = ff as Form of (VectQuot (V,(LKer f))),(VectQuot (W,(RKer (f *')))) ;
A12: CosetSet (V,(LKer f)) = the carrier of (VectQuot (V,(LKer f))) by VECTSP10:def 6;
A13: now :: thesis: for ww being Vector of (VectQuot (W,(RKer (f *')))) holds FunctionalSAF (ff,ww) is homogeneous
let ww be Vector of (VectQuot (W,(RKer (f *')))); :: thesis: FunctionalSAF (ff,ww) is homogeneous
consider w being Vector of W such that
A14: 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 F_Complex 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 F_Complex holds (FunctionalSAF (ff,ww)) . (r * A) = r * ((FunctionalSAF (ff,ww)) . A)
let r be Element of F_Complex; :: thesis: (FunctionalSAF (ff,ww)) . (r * A) = r * ((FunctionalSAF (ff,ww)) . A)
consider a being Vector of V such that
A15: A = a + (LKer f) by VECTSP10:22;
A16: ( the lmult of (VectQuot (V,(LKer f))) = lmultCoset (V,(LKer f)) & (lmultCoset (V,(LKer f))) . (r,A) = (r * a) + (LKer f) ) by A12, A15, VECTSP10:def 5, VECTSP10:def 6;
thus (FunctionalSAF (ff,ww)) . (r * A) = ff . (( the lmult of (VectQuot (V,(LKer f))) . (r,A)),ww) by BILINEAR:9
.= f . ((r * a),w) by A11, A14, A16
.= r * (f . (a,w)) by BILINEAR:31
.= r * (ff . (A,ww)) by A11, A14, A15
.= r * ((FunctionalSAF (ff,ww)) . A) by BILINEAR:9 ; :: thesis: verum
end;
hence FunctionalSAF (ff,ww) is homogeneous ; :: thesis: verum
end;
A17: CosetSet (W,(RKer (f *'))) = the carrier of (VectQuot (W,(RKer (f *')))) by VECTSP10:def 6;
A18: now :: thesis: for vv being Vector of (VectQuot (V,(LKer f))) holds FunctionalFAF (ff,vv) is cmplxhomogeneous
let vv be Vector of (VectQuot (V,(LKer f))); :: thesis: FunctionalFAF (ff,vv) is cmplxhomogeneous
consider v being Vector of V such that
A19: 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 F_Complex 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 F_Complex holds (FunctionalFAF (ff,vv)) . (r * A) = (r *') * ((FunctionalFAF (ff,vv)) . A)
let r be Element of F_Complex; :: thesis: (FunctionalFAF (ff,vv)) . (r * A) = (r *') * ((FunctionalFAF (ff,vv)) . A)
consider a being Vector of W such that
A20: A = a + (RKer (f *')) by VECTSP10:22;
A21: ( the lmult of (VectQuot (W,(RKer (f *')))) = lmultCoset (W,(RKer (f *'))) & (lmultCoset (W,(RKer (f *')))) . (r,A) = (r * a) + (RKer (f *')) ) by A17, A20, VECTSP10:def 5, VECTSP10:def 6;
thus (FunctionalFAF (ff,vv)) . (r * A) = ff . (vv,( the lmult of (VectQuot (W,(RKer (f *')))) . (r,A))) by BILINEAR:8
.= f . (v,(r * a)) by A11, A19, A21
.= (r *') * (f . (v,a)) by Th27
.= (r *') * (ff . (vv,A)) by A11, A19, A20
.= (r *') * ((FunctionalFAF (ff,vv)) . A) by BILINEAR:8 ; :: thesis: verum
end;
hence FunctionalFAF (ff,vv) is cmplxhomogeneous ; :: thesis: verum
end;
A22: 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
A23: 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)
let 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
A24: A = a + (LKer f) by VECTSP10:22;
consider b being Vector of V such that
A25: B = b + (LKer f) by VECTSP10:22;
A26: ( the addF of (VectQuot (V,(LKer f))) = addCoset (V,(LKer f)) & (addCoset (V,(LKer f))) . (A,B) = (a + b) + (LKer f) ) by A12, A24, A25, VECTSP10:def 3, VECTSP10:def 6;
thus (FunctionalSAF (ff,ww)) . (A + B) = ff . (( the addF of (VectQuot (V,(LKer f))) . (A,B)),ww) by BILINEAR:9
.= f . ((a + b),w) by A11, A23, A26
.= (f . (a,w)) + (f . (b,w)) by BILINEAR:26
.= (ff . (A,ww)) + (f . (b,w)) by A11, A23, A24
.= (ff . (A,ww)) + (ff . (B,ww)) by A11, A23, A25
.= ((FunctionalSAF (ff,ww)) . A) + (ff . (B,ww)) by BILINEAR:9
.= ((FunctionalSAF (ff,ww)) . A) + ((FunctionalSAF (ff,ww)) . B) by BILINEAR:9 ; :: thesis: verum
end;
hence FunctionalSAF (ff,ww) is additive ; :: thesis: verum
end;
now :: thesis: for vv being Vector of (VectQuot (V,(LKer f))) holds FunctionalFAF (ff,vv) is additive
let vv be Vector of (VectQuot (V,(LKer f))); :: thesis: FunctionalFAF (ff,vv) is additive
consider v being Vector of V such that
A27: 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)
let 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
A28: A = a + (RKer (f *')) by VECTSP10:22;
consider b being Vector of W such that
A29: B = b + (RKer (f *')) by VECTSP10:22;
A30: ( the addF of (VectQuot (W,(RKer (f *')))) = addCoset (W,(RKer (f *'))) & (addCoset (W,(RKer (f *')))) . (A,B) = (a + b) + (RKer (f *')) ) by A17, A28, A29, VECTSP10:def 3, VECTSP10:def 6;
thus (FunctionalFAF (ff,vv)) . (A + B) = ff . (vv,( the addF of (VectQuot (W,(RKer (f *')))) . (A,B))) by BILINEAR:8
.= f . (v,(a + b)) by A11, A27, A30
.= (f . (v,a)) + (f . (v,b)) by BILINEAR:27
.= (ff . (vv,A)) + (f . (v,b)) by A11, A27, A28
.= (ff . (vv,A)) + (ff . (vv,B)) by A11, A27, A29
.= ((FunctionalFAF (ff,vv)) . A) + (ff . (vv,B)) by BILINEAR:8
.= ((FunctionalFAF (ff,vv)) . A) + ((FunctionalFAF (ff,vv)) . B) by BILINEAR:8 ; :: thesis: verum
end;
hence FunctionalFAF (ff,vv) is additive ; :: thesis: verum
end;
then reconsider ff = ff as sesquilinear-Form of (VectQuot (V,(LKer f))),(VectQuot (W,(RKer (f *')))) by A22, A13, A18, Def4, BILINEAR:def 11, BILINEAR:def 12, BILINEAR:def 14;
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 A11; :: thesis: verum