let a be Element of INT.Ring; :: thesis: for V1, V2 being free finite-rank Z_Module
for f being Function of V1,V2
for b1 being OrdBasis of V1
for b2 being OrdBasis of V2 st a <> 0. INT.Ring holds
AutMt ((a * f),b1,b2) = a * (AutMt (f,b1,b2))
let V1, V2 be free finite-rank Z_Module; :: thesis: for f being Function of V1,V2
for b1 being OrdBasis of V1
for b2 being OrdBasis of V2 st a <> 0. INT.Ring holds
AutMt ((a * f),b1,b2) = a * (AutMt (f,b1,b2))
let f be Function of V1,V2; :: thesis: for b1 being OrdBasis of V1
for b2 being OrdBasis of V2 st a <> 0. INT.Ring holds
AutMt ((a * f),b1,b2) = a * (AutMt (f,b1,b2))
let b1 be OrdBasis of V1; :: thesis: for b2 being OrdBasis of V2 st a <> 0. INT.Ring holds
AutMt ((a * f),b1,b2) = a * (AutMt (f,b1,b2))
let b2 be OrdBasis of V2; :: thesis: ( a <> 0. INT.Ring implies AutMt ((a * f),b1,b2) = a * (AutMt (f,b1,b2)) )
assume A1: a <> 0. INT.Ring ; :: thesis: AutMt ((a * f),b1,b2) = a * (AutMt (f,b1,b2))
A2: width (AutMt ((a * f),b1,b2)) = width (AutMt (f,b1,b2))

proof

per cases ( len b1 = 0 or len b1 > 0 ) ;

suppose A3: len b1 = 0 ; :: thesis: width (AutMt ((a * f),b1,b2)) = width (AutMt (f,b1,b2))

then AutMt ((a * f),b1,b2) = {} by Th38
.= AutMt (f,b1,b2) by A3, Th38 ;
hence width (AutMt ((a * f),b1,b2)) = width (AutMt (f,b1,b2)) ; :: thesis: verum

end;

suppose A4: len b1 > 0 ; :: thesis: width (AutMt ((a * f),b1,b2)) = width (AutMt (f,b1,b2))

hence width (AutMt ((a * f),b1,b2)) = len b2 by Th39
.= width (AutMt (f,b1,b2)) by A4, Th39 ;
:: thesis: verum

end;

end;

end;

A5: width (AutMt ((a * f),b1,b2)) = width (a * (AutMt (f,b1,b2))) by A2, MATRIX_3:def 5;
A6: len (AutMt ((a * f),b1,b2)) = len b1 by Def8
.= len (AutMt (f,b1,b2)) by Def8 ;
A8: for i, j being Nat st [i,j] in Indices (AutMt ((a * f),b1,b2)) holds
(AutMt ((a * f),b1,b2)) * (i,j) = (a * (AutMt (f,b1,b2))) * (i,j)

proof

let i, j be Nat; :: thesis: ( [i,j] in Indices (AutMt ((a * f),b1,b2)) implies (AutMt ((a * f),b1,b2)) * (i,j) = (a * (AutMt (f,b1,b2))) * (i,j) )
assume A9: [i,j] in Indices (AutMt ((a * f),b1,b2)) ; :: thesis: (AutMt ((a * f),b1,b2)) * (i,j) = (a * (AutMt (f,b1,b2))) * (i,j)
then A11: [i,j] in Indices (AutMt (f,b1,b2)) by A2, A6, FINSEQ_3:29;
A11A: (AutMt ((a * f),b1,b2)) * (i,j) = a * ((AutMt (f,b1,b2)) * (i,j))

proof

consider p2 being FinSequence of INT such that
A12: p2 = (AutMt (f,b1,b2)) . i and
A13: (AutMt (f,b1,b2)) * (i,j) = p2 . j by A11, MATRIX_0:def 5;
A14: i in dom (AutMt ((a * f),b1,b2)) by A9, ZFMISC_1:87;
then A15: i in dom b1 by Lm3;
then i in dom (AutMt (f,b1,b2)) by Lm3;
then A16: p2 = (AutMt (f,b1,b2)) /. i by A12, PARTFUN1:def 6
.= (f . (b1 /. i)) |-- b2 by A15, Def8 ;
reconsider b4 = rng b2 as Basis of V2 by defOrdBasis;
consider p1 being FinSequence of INT such that
A17: p1 = (AutMt ((a * f),b1,b2)) . i and
A18: (AutMt ((a * f),b1,b2)) * (i,j) = p1 . j by A9, MATRIX_0:def 5;
consider KL1 being Linear_Combination of V2 such that
A19: ( (a * f) . (b1 /. i) = Sum KL1 & Carrier KL1 c= rng b2 ) and
A20: for t being Nat st 1 <= t & t <= len (((a * f) . (b1 /. i)) |-- b2) holds
(((a * f) . (b1 /. i)) |-- b2) /. t = KL1 . (b2 /. t) by Def7;
consider KL2 being Linear_Combination of V2 such that
A21: ( f . (b1 /. i) = Sum KL2 & Carrier KL2 c= rng b2 ) and
A22: for t being Nat st 1 <= t & t <= len ((f . (b1 /. i)) |-- b2) holds
((f . (b1 /. i)) |-- b2) /. t = KL2 . (b2 /. t) by Def7;
( b4 is linearly-independent & (a * f) . (b1 /. i) = a * (f . (b1 /. i)) ) by VECTSP_7:def 3, MATRLIN:def 4;
then A23: KL1 . (b2 /. j) = (a * KL2) . (b2 /. j) by A1, A19, A21, Th7
.= a * (KL2 . (b2 /. j)) by VECTSP_6:def 9 ;
A24: j in Seg (width (AutMt ((a * f),b1,b2))) by A9, ZFMISC_1:87;
then A25: 1 <= j by FINSEQ_1:1;
len b1 = len (AutMt ((a * f),b1,b2)) by Def8;
then dom b1 = dom (AutMt ((a * f),b1,b2)) by FINSEQ_3:29;
then dom b1 <> {} by A9;
then Seg (len b1) <> {} by FINSEQ_1:def 3;
then len b1 > 0 ;
then A26: j in Seg (len b2) by A24, Th39;
then A27: j <= len b2 by FINSEQ_1:1;
then j <= len ((f . (b1 /. i)) |-- b2) by Def7;
then A28: p2 /. j = KL2 . (b2 /. j) by A25, A16, A22;
A29: j in dom b2 by A26, FINSEQ_1:def 3;
then j in dom ((f . (b1 /. i)) |-- b2) by Lm1;
then A30: (AutMt (f,b1,b2)) * (i,j) = p2 /. j by A13, A16, PARTFUN1:def 6;
A31: p1 = (AutMt ((a * f),b1,b2)) /. i by A17, A14, PARTFUN1:def 6
.= ((a * f) . (b1 /. i)) |-- b2 by A15, Def8 ;
then A32: j in dom p1 by A29, Lm1;
j <= len (((a * f) . (b1 /. i)) |-- b2) by A27, Def7;
then p1 /. j = KL1 . (b2 /. j) by A25, A31, A20;
hence (AutMt ((a * f),b1,b2)) * (i,j) = a * ((AutMt (f,b1,b2)) * (i,j)) by A18, A32, A30, A28, A23, PARTFUN1:def 6; :: thesis: verum

end;

A110: [i,j] in Indices (AutMt (f,b1,b2)) by A2, A6, A9, FINSEQ_3:29;
(AutMt ((a * f),b1,b2)) * (i,j) = (a * (AutMt (f,b1,b2))) * (i,j) by A11A, A110, MATRIX_3:def 5;
hence (AutMt ((a * f),b1,b2)) * (i,j) = (a * (AutMt (f,b1,b2))) * (i,j) ; :: thesis: verum

end;

( len (AutMt ((a * f),b1,b2)) = len (AutMt ((a * f),b1,b2)) & len (AutMt ((a * f),b1,b2)) = len (a * (AutMt (f,b1,b2))) ) by A6, MATRIX_3:def 5;
hence AutMt ((a * f),b1,b2) = a * (AutMt (f,b1,b2)) by A5, A8, MATRIX_0:21; :: thesis: verum