theorem :: MESFUN17:69

for I, J, K being non empty closed_interval Subset of REAL
for f being PartFunc of [:[:RNS_Real,RNS_Real:],RNS_Real:],RNS_Real
for g being PartFunc of [:[:REAL,REAL:],REAL:],REAL
for Gzx being PartFunc of REAL,REAL st [:[:I,J:],K:] = dom f & f is_continuous_on [:[:I,J:],K:] & f = g & Gzx = (Integral1 (L-Meas,((Integral2 (L-Meas,(R_EAL g))) | [:I,J:]))) | J holds
( dom Gzx = J & Gzx is continuous & Gzx || J is bounded & Gzx is_integrable_on J & (Integral1 (L-Meas,((Integral2 (L-Meas,(R_EAL g))) | [:I,J:]))) | J is_integrable_on L-Meas & Integral (L-Meas,((Integral1 (L-Meas,((Integral2 (L-Meas,(R_EAL g))) | [:I,J:]))) | J)) = integral (Gzx,J) & Integral ((Prod_Measure (L-Meas,L-Meas)),((Integral2 (L-Meas,(R_EAL g))) | [:I,J:])) = integral (Gzx,J) )