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 Gxy being PartFunc of REAL,REAL st [:[:I,J:],K:] = dom f & f is_continuous_on [:[:I,J:],K:] & f = g & Gxy = (Integral1 ((Prod_Measure (L-Meas,L-Meas)),(R_EAL g))) | K holds
( Gxy || K is bounded & Gxy is_integrable_on K )
let f be PartFunc of [:[:RNS_Real,RNS_Real:],RNS_Real:],RNS_Real; :: thesis: for g being PartFunc of [:[:REAL,REAL:],REAL:],REAL
for Gxy being PartFunc of REAL,REAL st [:[:I,J:],K:] = dom f & f is_continuous_on [:[:I,J:],K:] & f = g & Gxy = (Integral1 ((Prod_Measure (L-Meas,L-Meas)),(R_EAL g))) | K holds
( Gxy || K is bounded & Gxy is_integrable_on K )
let g be PartFunc of [:[:REAL,REAL:],REAL:],REAL; :: thesis: for Gxy being PartFunc of REAL,REAL st [:[:I,J:],K:] = dom f & f is_continuous_on [:[:I,J:],K:] & f = g & Gxy = (Integral1 ((Prod_Measure (L-Meas,L-Meas)),(R_EAL g))) | K holds
( Gxy || K is bounded & Gxy is_integrable_on K )
let Gxy be PartFunc of REAL,REAL; :: thesis: ( [:[:I,J:],K:] = dom f & f is_continuous_on [:[:I,J:],K:] & f = g & Gxy = (Integral1 ((Prod_Measure (L-Meas,L-Meas)),(R_EAL g))) | K implies ( Gxy || K is bounded & Gxy is_integrable_on K ) )
assume that
A1: [:[:I,J:],K:] = dom f and
A2: f is_continuous_on [:[:I,J:],K:] and
A3: f = g and
A4: Gxy = (Integral1 ((Prod_Measure (L-Meas,L-Meas)),(R_EAL g))) | K ; :: thesis: ( Gxy || K is bounded & Gxy is_integrable_on K )
dom (Integral1 ((Prod_Measure (L-Meas,L-Meas)),(R_EAL g))) = REAL by FUNCT_2:def 1;
then dom Gxy = K by A4;
hence ( Gxy || K is bounded & Gxy is_integrable_on K ) by A1, A2, A3, A4, Th37, INTEGRA5:10, INTEGRA5:11; :: thesis: verum