
theorem Th4:
for A be non empty closed_interval Subset of REAL, f be Function of A,COMPLEX,
    D be Division of A,
    S be middle_volume of f,D
  holds
    Re S is middle_volume of (Re f),D
  & Im S is middle_volume of (Im f),D
proof
let A be non empty closed_interval Subset of REAL,
    f be Function of A,COMPLEX,
    D be Division of A,
    S be middle_volume of f,D;
A1: dom S = dom (Re S) by COMSEQ_3:def 3;
set RS = Re S;
    len S = len D by Def1; then
A2: dom S = Seg len D by FINSEQ_1:def 3; then
A3: len (RS) = len D by A1,FINSEQ_1:def 3;
  for i be Nat st i in dom D holds
    ex r be Element of REAL st r in rng ( (Re f)|divset(D,i))
                             & (RS).i=r*vol divset(D,i)
  proof
    let i be Nat such that A4: i in dom D;
    consider c be Element of COMPLEX such that
  A5: c in rng (f|divset(D,i)) & S.i= c * vol divset(D,i) by A4,Def1;
  A6: i in dom (Re S) by A4,A1,A2,FINSEQ_1:def 3;
    set r =Re c;
    take r;
    consider t be object such that
  A7: t in dom (f|divset(D,i)) and
  A8: c = (f|divset(D,i)).t by A5,FUNCT_1:def 3;
    t in dom(f) /\ divset(D,i) by A7,RELAT_1:61; then
    t in dom(f) by XBOOLE_0:def 4; then
  A9: t in dom (Re f) by COMSEQ_3:def 3;
  A10: dom (f|divset(D,i)) =dom (f) /\ divset(D,i) by RELAT_1:61
     .=dom (Re f) /\ divset(D,i) by COMSEQ_3:def 3
     .=dom ((Re f)|divset(D,i)) by RELAT_1:61;
    r = Re (f.t) by A7,A8,FUNCT_1:47
     .= (Re f).t by A9,COMSEQ_3:def 3
     .= ((Re f)|divset(D,i)).t by A7,A10,FUNCT_1:47;
    hence r in rng ( (Re f)|divset(D,i)) by A7,A10,FUNCT_1:def 3;
    thus (RS).i = Re (S.i) by A6,COMSEQ_3:def 3
               .= r*vol divset(D,i) by A5,Th1;
  end;
hence Re S is middle_volume of (Re f),D by A3,INTEGR15:def 1;
A11: dom S = dom (Im S) by COMSEQ_3:def 4;
set IS = Im S;
A12: len (IS) = len D by A2,A11,FINSEQ_1:def 3;
  for i be Nat st i in dom D holds
    ex r be Element of REAL st r in rng ( (Im f)|divset(D,i))
                             & (IS).i=r*vol divset(D,i)
  proof
    let i be Nat such that A13: i in dom D;
    consider c be Element of COMPLEX such that
  A14: c in rng (f|divset(D,i)) & S.i= c * vol divset(D,i) by A13,Def1;
  A15: i in dom (Im S) by A13,A2,A11,FINSEQ_1:def 3;
    set r = Im c;
    take r;
    consider t be object such that
  A16: t in dom (f|divset(D,i)) and
  A17: c = (f|divset(D,i)).t by A14,FUNCT_1:def 3;
    t in dom(f) /\ divset(D,i) by A16,RELAT_1:61; then
    t in dom(f) by XBOOLE_0:def 4; then
  A18: t in dom (Im f) by COMSEQ_3:def 4;
  A19: dom (f|divset(D,i)) =dom (f) /\ divset(D,i) by RELAT_1:61
     .=dom (Im f) /\ divset(D,i) by COMSEQ_3:def 4
     .=dom ((Im f)|divset(D,i)) by RELAT_1:61;
    r = Im (f.t) by A16,A17,FUNCT_1:47
     .= (Im f).t by A18,COMSEQ_3:def 4
     .= ((Im f)|divset(D,i)).t by A16,A19,FUNCT_1:47;
    hence r in rng ((Im f)|divset(D,i)) by A16,A19,FUNCT_1:def 3;
    thus (IS).i = Im (S.i) by A15,COMSEQ_3:def 4
               .= r*vol divset(D,i) by A14,Th1;
  end;
  hence Im S is middle_volume of (Im f),D by A12,INTEGR15:def 1;
end;
