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

theorem Th76:
  for D be non empty set, F be PartFunc of D,REAL, X,Y be set st
dom(F|(X \/ Y)) is finite & X misses Y holds FinS(F,X \/ Y), FinS(F,X) ^ FinS(F
  ,Y) are_fiberwise_equipotent
proof
  let D be non empty set, F be PartFunc of D,REAL, X,Y be set;
  assume
A1: dom(F|(X \/ Y)) is finite;
  F|Y c= F|(X \/ Y) by RELAT_1:75,XBOOLE_1:7;
  then reconsider dfy = dom(F|Y) as finite set by A1,FINSET_1:1,RELAT_1:11;
  defpred P[Nat] means for Y be set, Z being finite set st Z = dom(
F|Y) & dom(F|(X \/ Y)) is finite & X /\ Y = {} & $1 = card Z holds FinS(F,X \/
  Y), FinS(F,X) ^ FinS(F,Y) are_fiberwise_equipotent;
A2: card dfy = card dfy;
A3: for n st P[n] holds P[n+1]
  proof
    let n;
    assume
A4: P[n];
    let Y be set, Z be finite set;
    assume that
A5: Z = dom(F|Y) and
A6: dom(F|(X \/ Y)) is finite and
A7: X /\ Y = {} and
A8: n+1 = card Z;
    set x = the Element of dom(F|Y);
    reconsider x as Element of D by A5,A8,CARD_1:27,TARSKI:def 3;
    set y1 = Y\{x};
A9: dom(F|Y)= dom F /\ Y by RELAT_1:61;
    now
      assume
A10:  x in X;
      x in Y by A5,A8,A9,CARD_1:27,XBOOLE_0:def 4;
      hence contradiction by A7,A10,XBOOLE_0:def 4;
    end;
    then X \ {x} = X by ZFMISC_1:57;
    then
A11: (X \/ Y) \ {x} = X \/ y1 by XBOOLE_1:42;
A12: dom(F|y1) = dom F /\ y1 by RELAT_1:61;
A13: dom(F|y1) = dom(F|Y) \ {x}
    proof
      thus dom(F|y1) c= dom(F|Y) \ {x}
      proof
        let y be object;
        assume
A14:    y in dom(F|y1);
        then y in Y \ {x} by A12,XBOOLE_0:def 4;
        then
A15:    not y in {x} by XBOOLE_0:def 5;
        y in dom F by A12,A14,XBOOLE_0:def 4;
        then y in dom(F|Y) by A12,A9,A14,XBOOLE_0:def 4;
        hence thesis by A15,XBOOLE_0:def 5;
      end;
      let y be object;
      assume
A16:  y in dom(F|Y) \{x};
      then
A17:  not y in {x} by XBOOLE_0:def 5;
A18:  y in dom(F|Y) by A16,XBOOLE_0:def 5;
      then y in Y by A9,XBOOLE_0:def 4;
      then
A19:  y in y1 by A17,XBOOLE_0:def 5;
      y in dom F by A9,A18,XBOOLE_0:def 4;
      hence thesis by A12,A19,XBOOLE_0:def 4;
    end;
    then reconsider dFy = dom(F|y1) as finite set by A5;
    {x} c= dom(F|Y) by A5,A8,CARD_1:27,ZFMISC_1:31;
    then
A20: card dFy = n+1 - card {x} by A5,A8,A13,CARD_2:44
      .= n+1-1 by CARD_1:30
      .= n;
    X \/ y1 c= X \/ Y by XBOOLE_1:13;
    then dom F /\(X \/ y1) c= dom F /\ (X \/ Y) by XBOOLE_1:27;
    then dom(F|(X \/ y1)) c= dom F /\ (X \/ Y) by RELAT_1:61;
    then
A21: dom(F|(X \/ y1)) c= dom(F|(X \/ Y)) by RELAT_1:61;
A22: FinS(F,X \/ Y), F|(X \/ Y) are_fiberwise_equipotent by A6,Def13;
    dom(F|(X \/ Y)) = dom F /\ (X \/ Y) by RELAT_1:61
      .= dom F /\ X \/ dom F /\ Y by XBOOLE_1:23
      .= dom(F|X) \/ dom F /\ Y by RELAT_1:61
      .= dom(F|X) \/ dom(F|Y) by RELAT_1:61;
    then x in dom(F|(X \/ Y)) by A5,A8,CARD_1:27,XBOOLE_0:def 3;
    then FinS(F,X \/ y1)^<*F.x*>, F|(X \/ Y) are_fiberwise_equipotent by A6,A11
,Th66;
    then
A23: FinS(F,X \/ y1)^<*F.x*>, FinS(F,X \/ Y) are_fiberwise_equipotent by A22,
CLASSES1:76;
    X /\ y1 c= X /\ Y by XBOOLE_1:27;
    then FinS(F,X \/ y1), FinS(F,X) ^ FinS(F,y1) are_fiberwise_equipotent by A4
,A6,A7,A21,A20,XBOOLE_1:3;
    then FinS(F,X \/ y1) ^<*F.x*>, FinS(F,X) ^ FinS(F,y1) ^ <*F.x*>
    are_fiberwise_equipotent by RFINSEQ:1;
    then
A24: FinS(F,X \/ y1) ^<*F.x*>, FinS(F,X) ^ (FinS(F,y1) ^ <*F.x*>)
    are_fiberwise_equipotent by FINSEQ_1:32;
    FinS(F,y1)^<*F.x*>, F|Y are_fiberwise_equipotent & FinS(F,Y), F|Y
    are_fiberwise_equipotent by A5,A8,Def13,Th66,CARD_1:27;
    then FinS(F,y1)^<*F.x*>,FinS(F,Y) are_fiberwise_equipotent by CLASSES1:76;
    then
A25: FinS(F,y1)^<*F.x*> ^ FinS(F,X), FinS(F,Y) ^ FinS(F,X)
    are_fiberwise_equipotent by RFINSEQ:1;
    FinS(F,X)^(FinS(F,y1)^<*F.x*>),FinS(F,y1)^<*F.x*> ^ FinS(F,X)
    are_fiberwise_equipotent by RFINSEQ:2;
    then
    FinS(F,Y) ^ FinS(F,X), FinS(F,X) ^ FinS(F,Y) are_fiberwise_equipotent
& FinS (F,X)^(FinS(F,y1)^<*F.x*>), FinS(F,Y)^FinS(F,X) are_fiberwise_equipotent
    by A25,CLASSES1:76,RFINSEQ:2;
    then FinS(F,X)^(FinS(F,y1)^<*F.x*>), FinS(F,X)^FinS(F,Y)
    are_fiberwise_equipotent by CLASSES1:76;
    then FinS(F,X \/ y1) ^<*F.x*>, FinS(F,X) ^ FinS(F,Y)
    are_fiberwise_equipotent by A24,CLASSES1:76;
    hence thesis by A23,CLASSES1:76;
  end;
A26: P[ 0 ]
  proof
    let Y be set, Z be finite set;
    assume that
A27: Z = dom(F|Y) and
A28: dom(F|(X \/ Y)) is finite and
    X /\ Y = {} and
A29: 0 = card Z;
A30: dom(F|(X \/ Y)) = dom F /\ (X \/ Y) by RELAT_1:61
      .= dom F /\ X \/ dom F /\ Y by XBOOLE_1:23
      .= dom(F|X) \/ dom F /\ Y by RELAT_1:61
      .= dom(F|X) \/ dom(F|Y) by RELAT_1:61;
    then
A31: dom(F|X) is finite by A28,FINSET_1:1,XBOOLE_1:7;
A32: dom(F|Y) = {} by A27,A29;
    then FinS(F,X \/ Y) = FinS(F,dom(F|X)) by A28,A30,Th63
      .= FinS(F,X) by A31,Th63
      .= FinS(F,X)^<*>REAL by FINSEQ_1:34
      .= FinS(F,X)^FinS(F,dom(F|Y)) by A32,Th68
      .= FinS(F,X)^ FinS(F,Y) by A27,Th63;
    hence thesis;
  end;
A33: for n holds P[n] from NAT_1:sch 2(A26,A3);
  assume X /\ Y = {};
  hence thesis by A1,A33,A2;
end;
