reserve a,b,c for set;

theorem
  for D being non empty set,a,b be set,f,r,CR being File of D st a <> b
  & CR = <*b*> & f = <*b,a,b*> & r = <*a,b*> holds CR is_a_record_of f,CR & r
  is_a_record_of f,CR
proof
  let D be non empty set,a,b be set,f,r,CR be File of D;
  assume that
A1: a <> b and
A2: CR = <*b*> and
A3: f = <*b,a,b*> and
A4: r = <*a,b*>;
  reconsider b2=b,a2=a as Element of D by A3,Th10;
A5: CR.1=b & len CR=1 by A2,FINSEQ_1:40;
  f =<*b,a*>^<*b*> by A3,FINSEQ_1:43
    .=(f|2)^CR by A2,A3,Th14;
  then ovlpart(f,CR)=CR by FINSEQ_8:14;
  then CR/^(len ovlpart(f,CR))={} by FINSEQ_6:167;
  then f^(CR/^(len ovlpart(f,CR)))=f by FINSEQ_1:34;
  then
A6: ovlcon(f,CR)=f by FINSEQ_8:def 3;
A7: f=CR^r by A2,A3,A4,FINSEQ_1:43;
  then
A8: len f = len CR + len r by FINSEQ_1:22
    .= 1 + len <*a,b*> by A2,A4,FINSEQ_1:40
    .= 1 + 2 by FINSEQ_1:44
    .= 3;
  then CR^r is_substring_of f,1 by A7,FINSEQ_8:19;
  then
A9: CR^r is_substring_of addcr(f,CR),1 by A6,FINSEQ_8:def 11;
A10: len CR = 1 by A2,FINSEQ_1:40;
  then mid(f,1,len CR) = (f/^(1-'1))|(1-'1+1) by FINSEQ_6:def 3
    .= (f/^(1-'1))|(1+(0)) by NAT_2:8
    .= (f/^(0))|1 by NAT_D:34
    .= (f)|1
    .= (<*b2,a2,b2*>)|Seg 1 by A3,FINSEQ_1:def 16
    .= CR by A2,FINSEQ_6:4;
  then CR is_preposition_of f by A8,FINSEQ_8:def 8;
  then
A11: CR is_preposition_of addcr(f,CR) by A6,FINSEQ_8:def 11;
  r/^(1-'1)=r/^(0+1-'1) .=r/^0 by NAT_D:34
    .=r;
  then (r/^(1-'1)).1=a by A4;
  then
A12: not CR is_preposition_of (r/^(1-'1)) by A1,A5,FINSEQ_8:21;
A13: for j being Element of NAT st j >= 1 & j > 0 & CR is_preposition_of r/^
  (j-'1) holds j >= 2
  proof
    let j be Element of NAT;
    assume that
A14: j >= 1 and
    j > 0 and
A15: CR is_preposition_of r/^(j-'1);
    j>1 by A12,A14,A15,XXREAL_0:1;
    then 1+1<=j by NAT_1:13;
    hence thesis;
  end;
  r/^(2-'1) = r/^(1+1-'1) .= <*a2,b2*>/^1 by A4,NAT_D:34
    .= CR by A2,FINSEQ_6:46;
  then
A16: instr(1,r,CR) = 2 by A13,FINSEQ_8:def 10;
A17: len r = 2 by A4,FINSEQ_1:44;
  then len r + 1 -' len CR = 2 by A10,NAT_D:34;
  then CR is_terminated_by CR & r is_terminated_by CR by A10,A17,A16,
FINSEQ_8:28,def 12;
  hence thesis by A11,A9;
end;
