
theorem
  for D being non empty set,f,g being FinSequence of D
  st g is_postposition_of f holds
  len g>0 implies len g<=len f & mid(f,(len f+1) -' len g,len f)=g
proof
  let D be non empty set,f,g be FinSequence of D;
  assume
A1: g is_postposition_of f;
  then
A2: Rev g is_preposition_of Rev f;
  then
A3: len (Rev g)>0 implies 1<=len (Rev f) & mid(Rev f,1,len (Rev g))=Rev g;
A5: len (Rev f)=len f by FINSEQ_5:def 3;
  now
    assume
A6: len g>0;
    then
A7: 1<=len f by A2,FINSEQ_5:def 3;
A8: mid(Rev f,1,len g)=Rev g by A2,FINSEQ_5:def 3,A6;
A9: len f -1>=0 by A3,A5,A6,FINSEQ_5:def 3,XREAL_1:48;
A10: len g<=len f by A1,Th23;
A11: len f - len g>=0 by A1,Th23,XREAL_1:48;
    len f<len f+1 by XREAL_1:29;
    then len g<len f+1 by A10,XXREAL_0:2;
    then
A12: len f+1-len g>0 by XREAL_1:50;
A13: 0+1<=len g by A6,NAT_1:13;
A14: g=Rev Rev g
      .=mid(Rev (Rev f),len f-'len g+1,len f-'1+1)
        by A5,A7,A8,A10,A13,FINSEQ_6:113
      .=mid(f,len f -'len g +1,len f-'1+1);
A15: len f -'len g+1=len f-len g+1 by A11,XREAL_0:def 2
      .=(len f+1)-'len g by A12,XREAL_0:def 2;
    len f-'1+1=len f-1+1 by A9,XREAL_0:def 2
      .=len f;
    hence len g<=len f & mid(f,(len f+1) -' len g,len f)=g by A1,A14,A15,Th23;
  end;
  hence thesis;
end;
