reserve x,y for object,
        D,D1,D2 for non empty set,
        i,j,k,m,n for Nat,
        f,g for FinSequence of D*,
        f1 for FinSequence of D1*,
        f2 for FinSequence of D2*;
reserve f for complex-valued Function,
        g,h for complex-valued FinSequence;

theorem Th29:
  for f1,f2 be natural-valued Function st n>1 holds
    f1,f2 are_fiberwise_equipotent
  iff
    n|^f1,n|^f2 are_fiberwise_equipotent
proof
  let f1,f2 be natural-valued Function such that
  A1:n>1;
  set n1=n|^f1,n2=n|^f2;
  thus f1,f2 are_fiberwise_equipotent implies n|^f1,n|^f2
    are_fiberwise_equipotent
  proof
    assume A2:f1,f2 are_fiberwise_equipotent;
    for x be object holds card Coim(n1,x) = card Coim(n2,x)
    proof
      let x be object;
      A3:Coim(n1,x)= n1"{x} & Coim(n2,x)= n2"{x} by RELAT_1:def 17;
      A4:dom n1 = dom f1 & dom n2=dom f2 by Def4;
      per cases;
      suppose not x in rng n1 & not x in rng n2;
        then n1"{x}={} & n2"{x}={} by FUNCT_1:72;
        hence thesis by A3;
      end;
      suppose A5:x in rng n1 & not x in rng n2;
        then consider y be object such that
        A6: y in dom n1 & n1.y=x by FUNCT_1:def 3;
        A7: x = n to_power (f1.y) by A6,A4,Def4;
        f1.y in rng f1 by A6,A4,FUNCT_1:def 3;
        then f1"{f1.y}<>{} by FUNCT_1:72;
        then A8:Coim(f1,f1.y)<>{} by RELAT_1:def 17;
        card Coim(f1,f1.y) = card Coim(f2,f1.y) by A2,CLASSES1:def 10;
        then Coim(f2,f1.y)<>{} by A8;
        then f2"{f1.y}<>{} by RELAT_1:def 17;
        then f1.y in rng f2 by FUNCT_1:72;
        then consider z be object such that
        A9:z in dom f2 & f2.z = f1.y by FUNCT_1:def 3;
        A10:z in dom n2 by A9, Def4;
        n2.z = x by A9,Def4,A7;
        hence thesis by A10,FUNCT_1:def 3,A5;
      end;
      suppose A11:x in rng n2 & not x in rng n1;
        then consider y be object such that
        A12: y in dom n2 & n2.y=x by FUNCT_1:def 3;
        A13: x = n to_power (f2.y) by A12,A4,Def4;
        f2.y in rng f2 by A12,A4,FUNCT_1:def 3;
        then f2"{f2.y}<>{} by FUNCT_1:72;
        then A14:Coim(f2,f2.y)<>{} by RELAT_1:def 17;
        card Coim(f2,f2.y) = card Coim(f1,f2.y) by A2,CLASSES1:def 10;
        then Coim(f1,f2.y)<>{} by A14;
        then f1"{f2.y}<>{} by RELAT_1:def 17;
        then f2.y in rng f1 by FUNCT_1:72;
        then consider z be object such that
        A15: z in dom f1 & f1.z = f2.y by FUNCT_1:def 3;
        A16:z in dom n1 by A15,Def4;
        n1.z = x by A15,Def4,A13;
        hence thesis by A16,FUNCT_1:def 3,A11;
      end;
      suppose A17:x in rng n1 & x in rng n2;
        then consider y1 be object such that
        A18: y1 in dom n1 & n1.y1=x by FUNCT_1:def 3;
        A19: x = n to_power (f1.y1) by A18,A4,Def4;
        consider y2 be object such that
        A20: y2 in dom n2 & n2.y2=x by A17,FUNCT_1:def 3;
        A21: x = n to_power (f2.y2) by A20,A4,Def4;
        then A22: f2.y2 = f1.y1 by A19,A1,PEPIN:30;
        A23:Coim(f2,f2.y2) = Coim(n2,x) by A1,Th28, A21;
        Coim(f1,f1.y1) = Coim(n1,x) by A1,Th28,A19;
        hence thesis by A22,A2,CLASSES1:def 10,A23;
      end;
    end;
    hence thesis by CLASSES1:def 10;
  end;
  assume A24:n|^f1,n|^f2 are_fiberwise_equipotent;
  for x be object holds card Coim(f1,x) = card Coim(f2,x)
  proof
    let x be object;
    A25: Coim(f1,x) = f1"{x} & Coim(f2,x) = f2"{x} by RELAT_1:def 17;
    A26:dom n1 = dom f1 & dom n2=dom f2 by Def4;
    per cases;
    suppose not x in rng f1 & not x in rng f2;
      then f1"{x}={} & f2"{x}={} by FUNCT_1:72;
      hence thesis by A25;
    end;
    suppose A27:x in rng f1 & not x in rng f2;
      then consider y be object such that
      A28:y in dom f1 & f1.y=x by FUNCT_1:def 3;
      n1.y in rng n1 by A26,A28,FUNCT_1:def 3;
      then n1"{n1.y}<>{} by FUNCT_1:72;
      then A29:Coim(n1,n1.y)<>{} by RELAT_1:def 17;
      card Coim(n1,n1.y) = card Coim(n2,n1.y) by A24,CLASSES1:def 10;
      then Coim(n2,n1.y)<>{} by A29;
      then n2"{n1.y}<>{} by RELAT_1:def 17;
      then n1.y in rng n2 by FUNCT_1:72;
      then consider z be object such that
      A30: z in dom n2 & n2.z = n1.y by FUNCT_1:def 3;
      n2.z = n to_power (f2.z) & n1.y = n to_power (f1.y)
        by A28,A30,A26,Def4;
      then f2.z = f1.y by A30,A1,PEPIN:30;
      hence thesis by A30,A26,A28,FUNCT_1:def 3,A27;
    end;
    suppose A31:x in rng f2 & not x in rng f1;
      then consider y be object such that
      A32:y in dom f2 & f2.y=x by FUNCT_1:def 3;
      n2.y in rng n2 by A26,A32,FUNCT_1:def 3;
      then n2"{n2.y}<>{} by FUNCT_1:72;
      then A33:Coim(n2,n2.y)<>{} by RELAT_1:def 17;
      card Coim(n2,n2.y) = card Coim(n1,n2.y) by A24,CLASSES1:def 10;
      then Coim(n1,n2.y)<>{} by A33;
      then n1"{n2.y}<>{} by RELAT_1:def 17;
      then n2.y in rng n1 by FUNCT_1:72;
      then consider z be object such that
      A34: z in dom n1 & n1.z = n2.y by FUNCT_1:def 3;
      n1.z = n to_power (f1.z) & n2.y = n to_power (f2.y) by A32,A34,A26,Def4;
      then f1.z = f2.y by A34,A1,PEPIN:30;
      hence thesis by A34,A26,A32,FUNCT_1:def 3,A31;
    end;
    suppose A35:x in rng f1 & x in rng f2;
      then consider y1 be object such that
      A36: y1 in dom f1 & f1.y1=x by FUNCT_1:def 3;
      A37: n1.y1 = n to_power (f1.y1) by A36,Def4;
      consider y2 be object such that
      A38: y2 in dom f2 & f2.y2=x by A35,FUNCT_1:def 3;
      A39: n2.y2 = n to_power (f2.y2) by A38,Def4;
      then A40:card Coim(n2,n2.y2) = card Coim(n1,n1.y1)
        by A37,A38,A36,A24,CLASSES1:def 10;
      Coim(f2,f2.y2) = Coim(n2,n2.y2) by A1,Th28,A39;
      hence thesis by A1,Th28,A37,A40,A36,A38;
    end;
  end;
  hence thesis by CLASSES1:def 10;
end;
