let i, k, m, n be Nat; :: thesis:
let K be Field; :: thesis:
let a be Element of K; :: thesis:
let A9 be Matrix of m,n,K; :: thesis:
let B9 be Matrix of m,k,K; :: thesis:
assume A1: a <> 0. K ; :: thesis:
set RB = RLine (B9,i,(a * (Line (B9,i))));
set RA = RLine (A9,i,(a * (Line (A9,i))));
thus Solutions_of (A9,B9) c= Solutions_of ((RLine (A9,i,(a * (Line (A9,i))))),(RLine (B9,i,(a * (Line (B9,i)))))) :: according to XBOOLE_0:def 10 :: thesis:

proof

let x be object ; :: according to TARSKI:def 3 :: thesis:
assume A2: x in Solutions_of (A9,B9) ; :: thesis:
ex X being Matrix of K st
( x = X & len X = width A9 & width X = width B9 & A9 * X = B9 ) by A2;
hence x in Solutions_of ((RLine (A9,i,(a * (Line (A9,i))))),(RLine (B9,i,(a * (Line (B9,i)))))) by A2, Th38; :: thesis:

end;

let x be object ; :: according to TARSKI:def 3 :: thesis:
assume A3: x in Solutions_of ((RLine (A9,i,(a * (Line (A9,i))))),(RLine (B9,i,(a * (Line (B9,i)))))) ; :: thesis:

per cases ;

suppose A4: not i in Seg m ; :: thesis:

len A9 = m by MATRIX_0:def 2;
then ( len B9 = m & RLine (A9,i,(a * (Line (A9,i)))) = A9 ) by A4, MATRIX13:40, MATRIX_0:def 2;
hence x in Solutions_of (A9,B9) by A3, A4, MATRIX13:40; :: thesis:

end;

suppose A5: i in Seg m ; :: thesis:

reconsider aLA = a * (Line (A9,i)), aLB = a * (Line (B9,i)), aLAR = (a ") * (Line ((RLine (A9,i,(a * (Line (A9,i))))),i)), aLBR = (a ") * (Line ((RLine (B9,i,(a * (Line (B9,i))))),i)) as Element of the carrier of K * by FINSEQ_1:def 11;
set RRB = RLine ((RLine (B9,i,(a * (Line (B9,i))))),i,((a ") * (Line ((RLine (B9,i,(a * (Line (B9,i))))),i))));
set RRA = RLine ((RLine (A9,i,(a * (Line (A9,i))))),i,((a ") * (Line ((RLine (A9,i,(a * (Line (A9,i))))),i))));
A6: ex X being Matrix of K st
( x = X & len X = width (RLine (A9,i,(a * (Line (A9,i))))) & width X = width (RLine (B9,i,(a * (Line (B9,i))))) & (RLine (A9,i,(a * (Line (A9,i))))) * X = RLine (B9,i,(a * (Line (B9,i)))) ) by A3;
A7: len (a * (Line (A9,i))) = width A9 by CARD_1:def 7;
then A8: (a ") * (Line ((RLine (A9,i,(a * (Line (A9,i))))),i)) = (a ") * (a * (Line (A9,i))) by A5, MATRIX11:28
.= ((a ") * a) * (Line (A9,i)) by FVSUM_1:54
.= (1_ K) * (Line (A9,i)) by A1, VECTSP_1:def 10
.= Line (A9,i) by FVSUM_1:57 ;
A9: len (a * (Line (B9,i))) = width B9 by CARD_1:def 7;
then A10: (a ") * (Line ((RLine (B9,i,(a * (Line (B9,i))))),i)) = (a ") * (a * (Line (B9,i))) by A5, MATRIX11:28
.= ((a ") * a) * (Line (B9,i)) by FVSUM_1:54
.= (1_ K) * (Line (B9,i)) by A1, VECTSP_1:def 10
.= Line (B9,i) by FVSUM_1:57 ;
A11: width (RLine (B9,i,(a * (Line (B9,i))))) = width B9 by A9, MATRIX11:def 3;
A12: len ((a ") * (Line ((RLine (B9,i,(a * (Line (B9,i))))),i))) = width (RLine (B9,i,(a * (Line (B9,i))))) by CARD_1:def 7;
then A13: RLine ((RLine (B9,i,(a * (Line (B9,i))))),i,((a ") * (Line ((RLine (B9,i,(a * (Line (B9,i))))),i)))) = Replace ((RLine (B9,i,(a * (Line (B9,i))))),i,aLBR) by MATRIX11:29
.= Replace ((Replace (B9,i,aLB)),i,aLBR) by A9, MATRIX11:29
.= Replace (B9,i,aLBR) by FUNCT_7:34
.= RLine (B9,i,(Line (B9,i))) by A12, A11, A10, MATRIX11:29
.= B9 by MATRIX11:30 ;
A14: width (RLine (A9,i,(a * (Line (A9,i))))) = width A9 by A7, MATRIX11:def 3;
A15: len ((a ") * (Line ((RLine (A9,i,(a * (Line (A9,i))))),i))) = width (RLine (A9,i,(a * (Line (A9,i))))) by CARD_1:def 7;
then RLine ((RLine (A9,i,(a * (Line (A9,i))))),i,((a ") * (Line ((RLine (A9,i,(a * (Line (A9,i))))),i)))) = Replace ((RLine (A9,i,(a * (Line (A9,i))))),i,aLAR) by MATRIX11:29
.= Replace ((Replace (A9,i,aLA)),i,aLAR) by A7, MATRIX11:29
.= Replace (A9,i,aLAR) by FUNCT_7:34
.= RLine (A9,i,(Line (A9,i))) by A15, A14, A8, MATRIX11:29
.= A9 by MATRIX11:30 ;
hence x in Solutions_of (A9,B9) by A3, A6, A13, Th38; :: thesis:

end;