
theorem Th110:
  for X be non empty set, S be SigmaField of X, M be
sigma_Measure of S, f be PartFunc of X,ExtREAL, c be Real st f is_integrable_on
M holds c(#)f is_integrable_on M & Integral(M,c(#)f) =  c * Integral(M,f)
proof
  let X be non empty set, S be SigmaField of X, M be sigma_Measure of S, f be
  PartFunc of X,ExtREAL, c be Real such that
A1: f is_integrable_on M;
A2: integral+(M,max+f) <>+infty by A1;
  consider A be Element of S such that
A3: A = dom f and
A4: f is A-measurable by A1;
A5: c(#)f is A-measurable by A3,A4,Th49;
A6: dom(max-f) = A by A3,MESFUNC2:def 3;
A7: integral+(M,max-f) <>+infty by A1;
  0 <= integral+(M,max-f) by A1,Th96;
  then reconsider I2 = integral+(M,max-f) as Element of REAL
          by A7,XXREAL_0:14;
A8: max-f is nonnegative by Lm1;
  0 <= integral+(M,max+f) by A1,Th96;
  then reconsider I1 = integral+(M,max+f) as Element of REAL
           by A2,XXREAL_0:14;
A9: max+f is nonnegative by Lm1;
A10: dom(c(#)f) =A by A3,MESFUNC1:def 6;
A11: dom(max+f) = A by A3,MESFUNC2:def 2;
  per cases;
  suppose
A12: 0 <= c;
    c*I1 in REAL by XREAL_0:def 1;
    then
A13:  c * integral+(M,max+f) in REAL;
A14: max+(c(#)f)=c(#)max+f by A12,Th26;
    integral+(M,c(#)max+f) =  c * integral+(M,max+f) by A4,A9,A11,A12,Th86
,MESFUNC2:25;
    then
A15: integral+(M,max+(c(#)f)) < +infty by A14,A13,XXREAL_0:9;
    c*I2 in REAL by XREAL_0:def 1;
    then  c * integral+(M,max-f) is Element of REAL;
    then
A16:  c * integral+(M,max-f) < +infty by XXREAL_0:9;
A17: max-(c(#)f)=c(#)max-f by A12,Th26;
    integral+(M,c(#)max-f) =  c * integral+(M,max-f) by A3,A4,A8,A6,A12
,Th86,MESFUNC2:26;
    hence c(#)f is_integrable_on M by A5,A10,A17,A15,A16;
    thus Integral(M,c(#)f) =integral+(M,c(#)max+f) -integral+(M,max-(c(#)f))
    by A12,Th26
      .=integral+(M,c(#)max+f) -integral+(M,c(#)max-f) by A12,Th26
      .= c * integral+(M,max+f) - integral+(M,c(#)max-f) by A4,A9,A11,A12
,Th86,MESFUNC2:25
      .= c * integral+(M,max+f) -  c *integral+(M,max-f) by A3,A4,A8
,A6,A12,Th86,MESFUNC2:26
      .= c * Integral(M,f) by XXREAL_3:100;
  end;
  suppose
A18: c < 0;
    -(-c)=c;
    then consider a be Real such that
A19: c =-a and
A20: a > 0 by A18;
A21: max+(c(#)f)=a(#)max-f by A19,A20,Th27;
A22: max-(c(#)f)=a(#)max+f by A19,A20,Th27;
    a*I2 in REAL by XREAL_0:def 1;
    then
A23:  a *integral+(M,max-f) in REAL;
    integral+(M,a(#)max-f) =  a * integral+(M,max-f) by A3,A4,A8,A6,A20
,Th86,MESFUNC2:26;
    then
A24: integral+(M,max+(c(#)f)) < +infty by A21,A23,XXREAL_0:9;
    a*I1 in REAL by XREAL_0:def 1;
    then (a)*integral+(M,max+f) is Element of REAL;
    then
A25: (a)*integral+(M,max+f) < +infty by XXREAL_0:9;
    integral+(M,a(#)max+f) =  a * integral+(M,max+f) by A4,A9,A11,A20,Th86
,MESFUNC2:25;
    hence c(#)f is_integrable_on M by A5,A10,A22,A24,A25;
    thus Integral(M,c(#)f) = a * integral+(M,max-f) -integral+(M,a(#)max+
    f) by A3,A4,A8,A6,A20,A21,A22,Th86,MESFUNC2:26
      .= a * integral+(M,max-f)- a * integral+(M,max+f) by A4,A9,A11
,A20,Th86,MESFUNC2:25
      .= a * (integral+(M,max-f)-integral+(M,max+f)) by XXREAL_3:100
      .= a * (-(integral+(M,max+f)-integral+(M,max-f))) by XXREAL_3:26
      .=-( a * (integral+(M,max+f)-integral+(M,max-f))) by XXREAL_3:92
      .=  c * Integral(M,f) by A19,XXREAL_3:92;
  end;
end;
