theorem Th15:
  for f1 being FinSequence st
  k in dom f1 holds mid(f1,k,k)=<*f1.k*> & len mid(f1,k,k)=1
proof
  let f1 be FinSequence;
  assume
A0: k in dom f1; then
A1: 1<=k by FINSEQ_3:25;
A2: k<=len f1 by A0,FINSEQ_3:25;
  k-'1+1<=len f1 by A1,A2,XREAL_1:235;
  then
A4: k-'1+1-(k-'1)<=len f1-(k-'1) by XREAL_1:9;
  len (f1/^(k-'1))=len f1-'(k-'1) by RFINSEQ:29;
  then
A5: 1<=len (f1/^(k-'1)) by A4,NAT_D:39;
  k-'1+1=k by A1,XREAL_1:235;
  then
A6: (f1/^(k-'1)).1=f1.k by A2,Th113;
  k-'k+1=k-k+1 by XREAL_1:233
    .=1;
  then mid(f1,k,k)=(f1/^(k-'1))|1 by Def3
    .=<*(f1/^(k-'1)).1*> by A5,CARD_1:27,FINSEQ_5:20;
  hence thesis by A6,FINSEQ_1:39;
end;
