reserve i,j,m,n,k for Nat,
  x,y for set,
  K for Field,
  a,a1,a2 for Element of K,
  D for non empty set,
  d,d1,d2 for Element of D,
  M,M1,M2 for (Matrix of D),
  A,A1,A2,B1,B2 for (Matrix of K),
  f,g for FinSequence of NAT;
reserve F,F1,F2 for FinSequence_of_Matrix of D,
  G,G9,G1,G2 for FinSequence_of_Matrix of K;
reserve S,S1,S2 for FinSequence_of_Square-Matrix of D,
  R,R1,R2 for FinSequence_of_Square-Matrix of K;
reserve N for (Matrix of n,K),
  N1 for (Matrix of m,K);

theorem Th58:
  1.(K,f^g) = 1.(K,f)^1.(K,g)
proof
  set F=1.(K,f);
  set G=1.(K,g);
  set FG=F^G;
  set ONE=1.(K,f^g);
A1: len (f^g)=len f+len g by FINSEQ_1:22;
A2: dom ONE=dom (f^g) by Def8;
A3: dom G=dom g by Def8;
  then
A4: len G=len g by FINSEQ_3:29;
A5: dom F=dom f by Def8;
  then
A6: len F=len f by FINSEQ_3:29;
A7: len FG=len F+len G by FINSEQ_1:22;
A8: now
    let i such that
A9: 1<=i and
A10: i<= len FG;
    i in dom ONE by A2,A6,A4,A7,A1,A9,A10,FINSEQ_3:25;
    then
A11: ONE.i=1.(K,(f^g).i) by Def8;
A12: i in dom FG by A9,A10,FINSEQ_3:25;
    now
      per cases by A12,FINSEQ_1:25;
      suppose
A13:    i in dom F;
        hence ONE.i = 1.(K,f.i) by A5,A11,FINSEQ_1:def 7
          .= F.i by A13,Def8
          .= FG.i by A13,FINSEQ_1:def 7;
      end;
      suppose
        ex n st n in dom G & i=len F+n;
        then consider n such that
A14:    n in dom G and
A15:    i=len F+n;
        thus ONE.i = 1.(K,g.n) by A3,A6,A11,A14,A15,FINSEQ_1:def 7
          .= G.n by A14,Def8
          .= FG.i by A14,A15,FINSEQ_1:def 7;
      end;
    end;
    hence ONE.i=FG.i;
  end;
  len ONE=len (f^g) by A2,FINSEQ_3:29;
  hence thesis by A6,A4,A1,A8,FINSEQ_1:14,22;
end;
