reserve FT for non empty RelStr,
  A,B,C for Subset of FT;

theorem Th51:
  for g being FinSequence of FT,A being Subset of FT, x1,x2 being
Element of FT, k being Element of NAT st g is_minimum_path_in A,x1,x2 & 1<=k &
k<=len g holds g|k is continuous & rng (g|k) c=A & (g|k).1=x1 & (g|k).(len (g|k
  ))=g/.k
proof
  let g be FinSequence of FT,A be Subset of FT, x1,x2 be Element of FT, k be
  Element of NAT;
  assume that
A1: g is_minimum_path_in A,x1,x2 and
A2: 1<=k and
A3: k<=len g;
A4: k in dom g by A2,A3,FINSEQ_3:25;
  g is continuous by A1;
  hence g|k is continuous by A2,Th46;
A5: rng (g|k) c= rng g by FINSEQ_5:19;
  rng g c=A by A1;
  hence rng (g|k) c=A by A5;
  g.1=x1 by A1;
  hence (g|k).1=x1 by A2,FINSEQ_3:112;
  len (g|k)=k by A3,FINSEQ_1:59;
  hence (g|k).(len (g|k))=g.k by FINSEQ_3:112
    .=g/.k by A4,PARTFUN1:def 6;
end;
