theorem :: MESFUN17:68

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 Gzy being PartFunc of REAL,REAL st [:[:I,J:],K:] = dom f & f is_continuous_on [:[:I,J:],K:] & f = g & Gzy = (Integral2 (L-Meas,((Integral2 (L-Meas,(R_EAL g))) | [:I,J:]))) | I holds
( dom Gzy = I & Gzy is continuous & Gzy || I is bounded & Gzy is_integrable_on I & (Integral2 (L-Meas,((Integral2 (L-Meas,(R_EAL g))) | [:I,J:]))) | I is_integrable_on L-Meas & Integral (L-Meas,((Integral2 (L-Meas,((Integral2 (L-Meas,(R_EAL g))) | [:I,J:]))) | I)) = integral (Gzy,I) & Integral ((Prod_Measure (L-Meas,L-Meas)),((Integral2 (L-Meas,(R_EAL g))) | [:I,J:])) = integral (Gzy,I) )