
theorem Th22:
for m be Nat, M be Matrix of m,0,ExtREAL holds SumAll M = 0
proof
   let m be Nat, M be Matrix of m,0,ExtREAL;
   per cases;
   suppose m = 0; then
    len M = 0 by MATRIX_0:def 2;
    hence thesis by Th21;
   end;
   suppose m > 0; then
    len M > 0 by MATRIX_0:def 2; then
    reconsider k = len Sum M as non zero Nat by Def5;
    reconsider Z = 0. as R_eal;
    for k be Nat st k in dom Sum M holds (Sum M).k = 0
    proof
     len M = len Sum M by Def5; then
A2:  dom M = dom Sum M by FINSEQ_3:29;
     hereby let k be Nat;
      assume A3: k in dom Sum M; then
      M.k in rng M by A2,FUNCT_1:def 3; then
      M.k = <*>ExtREAL by MATRIX_0:def 2;
      hence (Sum M).k = 0 by A3,Def5,EXTREAL1:7;
     end;
    end; then
    Sum M = k |-> 0. by MATRPROB:1; then
    SumAll M = (len Sum M) * Z by Th20;
    hence SumAll M = 0;
   end;
end;
