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;

theorem Th63:
  i in Seg width G & width G = m+1 implies width DelCol(G,i) = m
proof
  set D = DelCol(G,i);
  assume that
A1: i in Seg width G and
A2: width G = m+1;
  width G <> 0 by A2;
  then 0<len G by Def3;
  then 0+1<=len G by NAT_1:13;
  then
 1 in dom G by FINSEQ_3:25;
  then
A3: Line(D,1) = Del(Line(G,1),i) by Th62;
A4: dom Line(G,1) = Seg len Line(G,1) & len Line(D,1) = width D by Def7,
FINSEQ_1:def 3;
  len Line(G,1) = m+1 by A2,Def7;
  hence thesis by A1,A2,A3,A4,FINSEQ_3:109;
end;
