let K be Field; :: thesis: for A, B being Matrix of st len A = len B & width A = 0 holds
( A ^^ B = B & B ^^ A = B )
let A, B be Matrix of ; :: thesis: ( len A = len B & width A = 0 implies ( A ^^ B = B & B ^^ A = B ) )
assume that
A1: len A = len B and
A2: width A = 0 ; :: thesis: ( A ^^ B = B & B ^^ A = B )
B2: Seg (width A) = {} by A1, A2;
set L = len A;
reconsider B' = B as Matrix of the carrier of K, len A, by A1, MATRIX_2:7;
reconsider A' = A as Matrix of the carrier of K, len A, by MATRIX_2:7;
set AB = A' ^^ B';
set BA = B' ^^ A';

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

suppose A3: len A = 0 ; :: thesis: ( A ^^ B = B & B ^^ A = B )

then len (B' ^^ A') = 0 by MATRIX_1:def 3;
then A4: B' ^^ A' = {} ;
len (A' ^^ B') = 0 by A3, MATRIX_1:def 3;
then A' ^^ B' = {} ;
hence ( A ^^ B = B & B ^^ A = B ) by A1, A3, A4; :: thesis: verum

end;

suppose A5: len A > 0 ; :: thesis: ( A ^^ B = B & B ^^ A = B )

then ( width (A' ^^ B') = width B & len (A' ^^ B') = len A ) by A2, MATRIX_1:24;
hence A ^^ B = Segm (A' ^^ B'),(Seg (len A)),((Seg (width B)) \ (Seg (width A))) by A2, FINSEQ_1:4, MATRIX13:46, B2
.= B by A2, Th19 ;
:: thesis: B ^^ A = B
( width (B' ^^ A') = width B & len (B' ^^ A') = len A ) by A2, A5, MATRIX_1:24;
hence B ^^ A = Segm (B' ^^ A'),(Seg (len A)),(Seg (width B)) by MATRIX13:46
.= B by Th19 ;
:: thesis: verum

end;

end;