let c, m, n be Nat; :: thesis: for D being non empty set
for AD being Matrix of n,m,D holds RCol (AD,c,(Col (AD,c))) = AD
let D be non empty set ; :: thesis: for AD being Matrix of n,m,D holds RCol (AD,c,(Col (AD,c))) = AD
let AD be Matrix of n,m,D; :: thesis: RCol (AD,c,(Col (AD,c))) = AD
set C = Col (AD,c);
set R = RCol (AD,c,(Col (AD,c)));

now :: thesis: for i, j being Nat st [i,j] in Indices AD holds
(RCol (AD,c,(Col (AD,c)))) * (i,j) = AD * (i,j)

reconsider c = c as Element of NAT by ORDINAL1:def 12;
let i, j be Nat; :: thesis: ( [i,j] in Indices AD implies (RCol (AD,c,(Col (AD,c)))) * (i,j) = AD * (i,j) )
assume A1: [i,j] in Indices AD ; :: thesis: (RCol (AD,c,(Col (AD,c)))) * (i,j) = AD * (i,j)
A2: len (Col (AD,c)) = len AD by MATRIX_0:def 8;
reconsider I = i, J = j as Element of NAT by ORDINAL1:def 12;
A3: i in dom AD by A1, ZFMISC_1:87;

now :: thesis: (RCol (AD,c,(Col (AD,c)))) * (i,j) = AD * (i,j)

per cases ( c = j or c <> J ) ;

suppose A4: c = j ; :: thesis: (RCol (AD,c,(Col (AD,c)))) * (i,j) = AD * (i,j)

hence (RCol (AD,c,(Col (AD,c)))) * (i,j) = (Col (AD,c)) . I by A1, A2, Def2
.= AD * (i,j) by A3, A4, MATRIX_0:def 8 ;
:: thesis: verum

end;

suppose c <> J ; :: thesis: (RCol (AD,c,(Col (AD,c)))) * (i,j) = AD * (i,j)

hence (RCol (AD,c,(Col (AD,c)))) * (i,j) = AD * (i,j) by A1, A2, Def2; :: thesis: verum

end;

end;

end;

hence (RCol (AD,c,(Col (AD,c)))) * (i,j) = AD * (i,j) ; :: thesis: verum

end;

hence RCol (AD,c,(Col (AD,c))) = AD by MATRIX_0:27; :: thesis: verum