let m, n be Nat; :: thesis:
let K be Field; :: thesis:
let M be Matrix of n,m,K; :: thesis:
let i, j be Nat; :: thesis:
let a be Element of K; :: thesis:
assume that
A1: M is without_repeated_line and
A2: j in dom M and
A3: ( i = j implies a <> - (1_ K) ) ; :: thesis:
A4: len M = n by MATRIX_0:def 2;
set L = (Line (M,i)) + (a * (Line (M,j)));
A5: dom M = Seg (len M) by FINSEQ_1:def 3;
set R = RLine (M,i,((Line (M,i)) + (a * (Line (M,j)))));

hence Lin (lines M) = Lin (lines (RLine (M,i,((Line (M,i)) + (a * (Line (M,j))))))) by A5, MATRIX13:40; :: thesis:

end;

then n <> 0 by A5, A4;
then A7: width M = m by MATRIX_0:23;
then reconsider Li = Line (M,i), Lj = Line (M,j) as Vector of (m -VectSp_over K) by MATRIX13:102;
a * Lj = a * (Line (M,j)) by A7, MATRIX13:102;
then A8: (Line (M,i)) + (a * (Line (M,j))) = Li + (a * Lj) by A7, MATRIX13:102;
A9: ( not Li = Lj or a <> - (1_ K) or Li = 0. (m -VectSp_over K) )

end;

reconsider L9 = (Line (M,i)) + (a * (Line (M,j))) as Element of the carrier of K * by FINSEQ_1:def 11;
reconsider LL = L9 as set ;
set iL = {i} --> L9;
len ((Line (M,i)) + (a * (Line (M,j)))) = width M by CARD_1:def 7;
then A11: RLine (M,i,((Line (M,i)) + (a * (Line (M,j))))) = M +* (i,LL) by MATRIX11:29
.= M +* (i .--> LL) by A6, FUNCT_7:def 3
.= M +* ({i} --> L9) by FUNCOP_1:def 9 ;
M .: ((dom M) \ (dom ({i} --> L9))) = (M .: (dom M)) \ (M .: (dom ({i} --> L9))) by A1, FUNCT_1:64
.= (rng M) \ (M .: (dom ({i} --> L9))) by RELAT_1:113
.= (rng M) \ (Im (M,i))
.= (rng M) \ {(M . i)} by A6, FUNCT_1:59
.= (rng M) \ {(Line (M,i))} by A5, A4, A6, MATRIX_0:52 ;
then A12: lines (RLine (M,i,((Line (M,i)) + (a * (Line (M,j)))))) = ((lines M) \ {(Line (M,i))}) \/ (rng ({i} --> L9)) by A11, FRECHET:12
.= ((lines M) \ {(Line (M,i))}) \/ {((Line (M,i)) + (a * (Line (M,j))))} by FUNCOP_1:8 ;
A13: Lj in lines M by A2, A5, A4, MATRIX13:103;
Li in lines M by A5, A4, A6, MATRIX13:103;
hence Lin (lines M) = Lin (lines (RLine (M,i,((Line (M,i)) + (a * (Line (M,j))))))) by A8, A12, A13, A9, Th14; :: thesis:

end;