
theorem Th27:
for p be ext-real number, M be Matrix of ExtREAL st
 (for i be Nat st i in dom M holds not p in rng (M.i)) holds
  (for j be Nat st j in dom (M@) holds not p in rng((M@).j))
proof
   let p be ext-real number;
   let M be Matrix of ExtREAL;
   assume
A1: for i be Nat st i in dom M holds not p in rng (M.i);
   hereby let j be Nat;
    assume A2: j in dom(M@); then
A3: M@.j = Line(M@,j) by MATRIX_0:60;
    j in Seg len (M@) by A2,FINSEQ_1:def 3; then
    j in Seg width M by MATRIX_0:def 6; then
A5: Line(M@,j) = Col(M,j) by MATRIX_0:59;

    for v be object st v in dom Line(M@,j) holds Line(M@,j).v <> p
    proof
     let v be object;
     assume A6: v in dom Line(M@,j); then
     reconsider i = v as Element of NAT;
     1 <= i & i <= len Line(M@,j) by A6,FINSEQ_3:25; then
     1 <= i & i <= width (M@) by MATRIX_0:def 7; then
     i in Seg width (M@); then
     [j,i] in [:dom(M@),Seg width(M@):] by A2,ZFMISC_1:def 2; then
     [j,i] in Indices (M@) by MATRIX_0:def 4; then
A7:  [i,j] in Indices M by MATRIX_0:def 6; then
A8:  i in dom M & j in dom (M.i) by MATRPROB:13; then
     Line(M@,j).v = M*(i,j) by A5,MATRIX_0:def 8; then
     Line(M@,j).v = (M.i).j by A7,MATRPROB:14; then
     Line(M@,j).v in rng (M.i) by A8,FUNCT_1:3;
     hence Line(M@,j).v <> p by A1,A7,MATRPROB:13;
    end;
    hence not p in rng (M@.j) by A3,FUNCT_1:def 3;
   end;
end;
