reserve a,b,c for set;

theorem Th5:
  for D being non empty set,f,g being FinSequence of D st len g>=1
  holds mid(f^g,len f+1,len f+len g)=g
proof
  let D be non empty set,f,g be FinSequence of D;
  assume
A1: len g>=1;
A2: g|len g=g|(Seg len g) by FINSEQ_1:def 16;
  per cases;
  suppose
A3: len f+1<=len f+len g;
    then
A4: len f < len f+len g by NAT_1:13;
    then len f-len f < len f+len g-len f by XREAL_1:14;
    then
A5: len f+len g-'len f=len g by XREAL_0:def 2;
    mid(f^g,len f+1,len f+len g)=((f^g)/^(len f+1-'1))|(len f+len g-'(len
    f+1) + 1 ) by A3,FINSEQ_6:def 3
      .=((f^g)/^len f)|(len f+len g-'(len f+1)+1) by NAT_D:34
      .=((f^g)/^len f)|(len f+len g-'len f) by A4,NAT_2:7
      .=g|len g by A5;
    hence thesis by A2,FINSEQ_2:20;
  end;
  suppose
    len f+1>len f+len g;
    then len f+1-len f>len f+len g-len f by XREAL_1:14;
    hence thesis by A1;
  end;
end;
