reserve x,y,z for object,
  i,j,n,m for Nat,
  D for non empty set,
  s,t for FinSequence,
  a,a1,a2,b1,b2,d for Element of D,
  p, p1,p2,q,r for FinSequence of D;
reserve M,M1,M2 for Matrix of D;
reserve f for FinSequence of D;
reserve i,j,i1,j1 for Nat;
reserve k for Nat, G for Matrix of D;
reserve x,y,x1,x2,y1,y2 for object,
  i,j,k,l,n,m for Nat,
  D for non empty set,
  s,s2 for FinSequence,
  a,b,c,d for Element of D,
  q,r for FinSequence of D,
  a9,b9 for Element of D;
reserve m for Nat;
reserve f for FinSequence of D,
        G for Matrix of D;

theorem
  rng f misses rng Col(G,i) & f/.n in rng Line(G,m) & n in dom f & i in
  Seg width G & m in dom G & width G>1 implies f/.n in rng Line(DelCol(G,i),m)
proof
  set D = DelCol(G,i);
  assume that
A1: rng f misses rng Col(G,i) and
A2: f/.n in rng Line(G,m) and
A3: n in dom f and
A4: i in Seg width G and
A5: m in dom G and
A6: width G>1;
A7: len Line(D,m)=width D & dom Line(D,m) = Seg len Line(D,m) by Def7,
FINSEQ_1:def 3;
  consider j being Nat such that
A8: j in dom Line(G,m) and
A9: f/.n=Line(G,m).j by A2,FINSEQ_2:10;
  reconsider j as Nat;
A10: len Line(G,m) = width G by Def7;
  then
A11: j<=width G by A8,FINSEQ_3:25;
A12: dom Line(G,m) = Seg len Line(G,m) by FINSEQ_1:def 3;
  then
A13: 1<=i by A4,A10,FINSEQ_3:25;
A14: f/.n=G*(m,j) by A8,A9,A12,A10,Def7;
A15: i<=width G by A4,A12,A10,FINSEQ_3:25;
A16: 1<=j by A8,FINSEQ_3:25;
  consider M be Nat such that
A17: width G = M+1 and
A18: M>0 by A6,SEQM_3:43;
A19: width D = M by A4,A17,Th63;
  i <> j by A1,A3,A5,A14,Th43;
  then per cases by XXREAL_0:1;
  suppose
A20: j<i;
    then j<width G by A15,XXREAL_0:2;
    then j<=M by A17,NAT_1:13;
    then
A21: j in Seg width D by A16,A19,FINSEQ_1:1;
    f/.n=D*(m,j) by A4,A5,A14,A16,A17,A18,A20,Th69;
    then Line(D,m).j=f/.n by A21,Def7;
    hence thesis by A7,A21,FUNCT_1:def 3;
  end;
  suppose
A22: i<j;
    reconsider l=j-1 as Element of NAT by A16,INT_1:5;
A23: l<=M by A11,A17,XREAL_1:20;
    i+1<=j by A22,NAT_1:13;
    then
A24: i<=l by XREAL_1:19;
    then 1<=l by A13,XXREAL_0:2;
    then
A25: l in Seg width D by A19,A23,FINSEQ_1:1;
    l+1=j;
    then f/.n=D*(m,l) by A4,A5,A14,A13,A17,A24,A23,Th70;
    then Line(D,m).l=f/.n by A25,Def7;
    hence thesis by A7,A25,FUNCT_1:def 3;
  end;
end;
