let I, J, K be non empty closed_interval Subset of REAL; :: thesis: 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) )
let f be PartFunc of [:[:RNS_Real,RNS_Real:],RNS_Real:],RNS_Real; :: thesis: 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) )
let g be PartFunc of [:[:REAL,REAL:],REAL:],REAL; :: thesis: 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) )
let Gzx be PartFunc of REAL,REAL; :: thesis: ( [:[: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 implies ( 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) ) )
assume that
A1: [:[:I,J:],K:] = dom f and
A2: f is_continuous_on [:[:I,J:],K:] and
A3: f = g and
A4: Gzx = (Integral1 (L-Meas,((Integral2 (L-Meas,(R_EAL g))) | [:I,J:]))) | J ; :: thesis: ( 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) )
A5: dom (Integral2 (L-Meas,(R_EAL g))) = [:REAL,REAL:] by FUNCT_2:def 1;
reconsider Gz = (Integral2 (L-Meas,(R_EAL g))) | [:I,J:] as PartFunc of [:REAL,REAL:],REAL by A1, A2, A3, Th32;
reconsider Fz = Gz as PartFunc of [:RNS_Real,RNS_Real:],RNS_Real ;
A6: Fz is_uniformly_continuous_on [:I,J:] by A1, A2, A3, Th34;
then A7: Fz is_continuous_on [:I,J:] by NFCONT_2:7;
A8: Gzx = (Integral1 (L-Meas,(R_EAL Gz))) | J by A4, MESFUNC5:def 7;
A9: dom (Integral1 (L-Meas,(R_EAL Gz))) = REAL by FUNCT_2:def 1;
hence A10: dom Gzx = J by A8; :: thesis: ( 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) )
then A11: ( Gzx || J is bounded & Gzx is_integrable_on J & Integral ((Prod_Measure (L-Meas,L-Meas)),(R_EAL Gz)) = integral (Gzx,J) ) by A7, A5, A8, MESFUN16:56, MESFUN16:59, INTEGRA5:10, INTEGRA5:11;
thus Gzx is continuous by A6, A5, A8, NFCONT_2:7, MESFUN16:56; :: thesis: ( 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) )
hence ( Gzx || J is bounded & Gzx is_integrable_on J ) by A9, A8, INTEGRA5:10, INTEGRA5:11; :: thesis: ( (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) )
( I in L-Field & J in L-Field ) by MEASUR10:5, MEASUR12:75;
then ( Gzx is_integrable_on L-Meas & Integral (L-Meas,(Gzx | J)) = integral (Gzx,J) ) by A10, A11, MESFUN14:49;
hence ( (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) ) by A4, MESFUNC5:def 7; :: thesis: Integral ((Prod_Measure (L-Meas,L-Meas)),((Integral2 (L-Meas,(R_EAL g))) | [:I,J:])) = integral (Gzx,J)
thus Integral ((Prod_Measure (L-Meas,L-Meas)),((Integral2 (L-Meas,(R_EAL g))) | [:I,J:])) = integral (Gzx,J) by A11, MESFUNC5:def 7; :: thesis: verum