reserve n,m for Nat,
  r,r1,r2,s,t for Real,
  x,y for set;

theorem Th70:
  for D be non empty set, F be PartFunc of D,REAL, X be set, d be
Element of D st dom(F|X) is finite & d in dom(F|X) & FinS(F,X).(len FinS(F,X))
  = F.d holds FinS(F,X) = FinS(F,X\{d}) ^ <*F.d*>
proof
  let D be non empty set, F be PartFunc of D,REAL, X be set, d be Element of D;
  set dx = dom(F|X), fx = FinS(F,X), fy = FinS(F,X \{d});
  assume that
A1: dx is finite and
A2: d in dx and
A3: fx.(len fx) = F.d;
A4: fx, F|X are_fiberwise_equipotent by A1,Def13;
  then rng fx = rng(F|X) by CLASSES1:75;
  then fx <> {} by A2,FUNCT_1:3,RELAT_1:38;
  then 0+1<=len fx by NAT_1:13;
  then max(0,len fx -1) = len fx - 1 by FINSEQ_2:4;
  then reconsider n=len fx - 1 as Element of NAT by FINSEQ_2:5;
  len fx = n+1;
  then
A5: fx = fx|n ^ <*F.d*> by A3,RFINSEQ:7;
A6: fx|n is non-increasing by RFINSEQ:20;
  fy ^ <*F.d*>, F|X are_fiberwise_equipotent by A1,A2,Th66;
  then fx, fy^<*F.d*> are_fiberwise_equipotent by A4,CLASSES1:76;
  then fy, fx|n are_fiberwise_equipotent by A5,RFINSEQ:1;
  hence thesis by A5,A6,RFINSEQ:23;
end;
