
theorem Th11:
  for D being non empty set,f,g being FinSequence of D
  holds ovlcon(f,g)=(f|(len f-'len ovlpart(f,g)))^g
proof
  let D be non empty set,f,g be FinSequence of D;
A1: len f-'len ovlpart(f,g)=len f-len ovlpart(f,g) by Th10,XREAL_1:233;
  len f+1<=len f+1+len ovlpart(f,g) by NAT_1:12;
  then len f+1-len ovlpart(f,g)<= len f+1+len ovlpart(f,g)-len ovlpart(f,g)
  by XREAL_1:9;
  then
A2: len f+1-'(len f-'len ovlpart(f,g)+1) =len f +1-(len f-'len ovlpart(f
  ,g)+1) by A1,XREAL_1:233
    .= len ovlpart(f,g) by A1;
  len f-'len ovlpart(f,g)<=len f by NAT_D:35;
  then
A3: len (f/^(len f-'len ovlpart(f,g)))
  =len f-(len f-'len ovlpart(f,g)) by RFINSEQ:def 1
    .=0+len ovlpart(f,g) by A1;
A4: ovlpart(f,g)=smid(f,len f-'len ovlpart(f,g)+1,len f) by Def2
    .=(f/^(len f-'len ovlpart(f,g)))|(len ovlpart(f,g)) by A2,NAT_D:34
    .=(f/^(len f-'len ovlpart(f,g))) by A3,FINSEQ_1:58;
  ovlpart(f,g)=smid(g,1,len ovlpart(f,g)) by Def2
    .=(g/^(0+1-'1))|(len ovlpart(f,g)) by NAT_D:34
    .=(g/^(0))|(len ovlpart(f,g)) by NAT_D:34
    .= g|(len ovlpart(f,g)) by FINSEQ_5:28;
  hence ovlcon(f,g)= (f|(len f-'len ovlpart(f,g)))^
  (g|(len ovlpart(f,g)))^(g/^(len ovlpart(f,g))) by A4,RFINSEQ:8
    .= (f|(len f-'len ovlpart(f,g)))^
  ((g|(len ovlpart(f,g)))^(g/^(len ovlpart(f,g)))) by FINSEQ_1:32
    .=(f|(len f-'len ovlpart(f,g)))^g by RFINSEQ:8;
end;
