let n be non zero Element of NAT ; :: thesis: for I being Function of REAL,(REAL-NS 1)
for x0 being Point of (REAL-NS 1)
for y0 being Element of REAL
for g being PartFunc of REAL,(REAL-NS n)
for f being PartFunc of (REAL-NS 1),(REAL-NS n) st I = (proj (1,1)) " & x0 in dom f & y0 in dom g & x0 = <*y0*> & f * I = g holds
( f is_continuous_in x0 iff g is_continuous_in y0 )
let I be Function of REAL,(REAL-NS 1); :: thesis: for x0 being Point of (REAL-NS 1)
for y0 being Element of REAL
for g being PartFunc of REAL,(REAL-NS n)
for f being PartFunc of (REAL-NS 1),(REAL-NS n) st I = (proj (1,1)) " & x0 in dom f & y0 in dom g & x0 = <*y0*> & f * I = g holds
( f is_continuous_in x0 iff g is_continuous_in y0 )
let x0 be Point of (REAL-NS 1); :: thesis: for y0 being Element of REAL
for g being PartFunc of REAL,(REAL-NS n)
for f being PartFunc of (REAL-NS 1),(REAL-NS n) st I = (proj (1,1)) " & x0 in dom f & y0 in dom g & x0 = <*y0*> & f * I = g holds
( f is_continuous_in x0 iff g is_continuous_in y0 )
let y0 be Element of REAL ; :: thesis: for g being PartFunc of REAL,(REAL-NS n)
for f being PartFunc of (REAL-NS 1),(REAL-NS n) st I = (proj (1,1)) " & x0 in dom f & y0 in dom g & x0 = <*y0*> & f * I = g holds
( f is_continuous_in x0 iff g is_continuous_in y0 )
let g be PartFunc of REAL,(REAL-NS n); :: thesis: for f being PartFunc of (REAL-NS 1),(REAL-NS n) st I = (proj (1,1)) " & x0 in dom f & y0 in dom g & x0 = <*y0*> & f * I = g holds
( f is_continuous_in x0 iff g is_continuous_in y0 )
let f be PartFunc of (REAL-NS 1),(REAL-NS n); :: thesis: ( I = (proj (1,1)) " & x0 in dom f & y0 in dom g & x0 = <*y0*> & f * I = g implies ( f is_continuous_in x0 iff g is_continuous_in y0 ) )
assume A1: ( I = (proj (1,1)) " & x0 in dom f & y0 in dom g & x0 = <*y0*> & f * I = g ) ; :: thesis: ( f is_continuous_in x0 iff g is_continuous_in y0 )
reconsider J = proj (1,1) as Function of (REAL-NS 1),REAL by Lm1;
thus ( f is_continuous_in x0 implies g is_continuous_in y0 ) :: thesis: ( g is_continuous_in y0 implies f is_continuous_in x0 )

proof

I /. y0 = x0 by A1, PDIFF_1:1;
hence ( f is_continuous_in x0 implies g is_continuous_in y0 ) by A1, Th33, NFCONT_3:15; :: thesis: verum

end;

A2: I * J = id (REAL-NS 1) by A1, Lm2, Lm1, FUNCT_1:39;
A3: g * J = f * (id (REAL-NS 1)) by A2, A1, RELAT_1:36
.= f by FUNCT_2:17 ;
J /. x0 = y0 by A1, PDIFF_1:1;
hence ( g is_continuous_in y0 implies f is_continuous_in x0 ) by A3, Th32, A1, Th34; :: thesis: verum