hereby :: thesis: ( ( for o being Object of C holds
( <^o,o^> <> {} & ex i being Morphism of o,o st
for o9 being Object of C
for m9 being Morphism of o9,o
for m99 being Morphism of o,o9 holds
( ( <^o9,o^> <> {} implies i * m9 = m9 ) & ( <^o,o9^> <> {} implies m99 * i = m99 ) ) ) ) implies C is with_units )
assume A1: C is with_units ; :: thesis: for o being Object of C holds
( <^o,o^> <> {} & ex i being Morphism of o,o st
for o9 being Object of C
for m9 being Morphism of o9,o
for m99 being Morphism of o,o9 holds
( ( <^o9,o^> <> {} implies i * m9 = m9 ) & ( <^o,o9^> <> {} implies m99 * i = m99 ) ) )

then reconsider C9 = C as non empty with_units AltCatStr ;
let o be Object of C; :: thesis: ( <^o,o^> <> {} & ex i being Morphism of o,o st
for o9 being Object of C
for m9 being Morphism of o9,o
for m99 being Morphism of o,o9 holds
( ( <^o9,o^> <> {} implies i * m9 = m9 ) & ( <^o,o9^> <> {} implies m99 * i = m99 ) ) )

thus <^o,o^> <> {} by ; :: thesis: ex i being Morphism of o,o st
for o9 being Object of C
for m9 being Morphism of o9,o
for m99 being Morphism of o,o9 holds
( ( <^o9,o^> <> {} implies i * m9 = m9 ) & ( <^o,o9^> <> {} implies m99 * i = m99 ) )

reconsider p = o as Object of C9 ;
reconsider i = idm p as Morphism of o,o ;
take i = i; :: thesis: for o9 being Object of C
for m9 being Morphism of o9,o
for m99 being Morphism of o,o9 holds
( ( <^o9,o^> <> {} implies i * m9 = m9 ) & ( <^o,o9^> <> {} implies m99 * i = m99 ) )

let o9 be Object of C; :: thesis: for m9 being Morphism of o9,o
for m99 being Morphism of o,o9 holds
( ( <^o9,o^> <> {} implies i * m9 = m9 ) & ( <^o,o9^> <> {} implies m99 * i = m99 ) )

let m9 be Morphism of o9,o; :: thesis: for m99 being Morphism of o,o9 holds
( ( <^o9,o^> <> {} implies i * m9 = m9 ) & ( <^o,o9^> <> {} implies m99 * i = m99 ) )

let m99 be Morphism of o,o9; :: thesis: ( ( <^o9,o^> <> {} implies i * m9 = m9 ) & ( <^o,o9^> <> {} implies m99 * i = m99 ) )
thus ( <^o9,o^> <> {} implies i * m9 = m9 ) by ALTCAT_1:20; :: thesis: ( <^o,o9^> <> {} implies m99 * i = m99 )
thus ( <^o,o9^> <> {} implies m99 * i = m99 ) by ALTCAT_1:def 17; :: thesis: verum
end;
assume A2: for o being Object of C holds
( <^o,o^> <> {} & ex i being Morphism of o,o st
for o9 being Object of C
for m9 being Morphism of o9,o
for m99 being Morphism of o,o9 holds
( ( <^o9,o^> <> {} implies i * m9 = m9 ) & ( <^o,o9^> <> {} implies m99 * i = m99 ) ) ) ; :: thesis: C is with_units
hereby :: according to ALTCAT_1:def 7,ALTCAT_1:def 16 :: thesis: the Comp of C is with_right_units
let j be Element of C; :: thesis: ex e being set st
( e in the Arrows of C . (j,j) & ( for i being Element of C
for f being set st f in the Arrows of C . (i,j) holds
( the Comp of C . (i,j,j)) . (e,f) = f ) )

reconsider o = j as Object of C ;
consider m being Morphism of o,o such that
A3: for o9 being Object of C
for m9 being Morphism of o9,o
for m99 being Morphism of o,o9 holds
( ( <^o9,o^> <> {} implies m * m9 = m9 ) & ( <^o,o9^> <> {} implies m99 * m = m99 ) ) by A2;
reconsider e = m as set ;
take e = e; :: thesis: ( e in the Arrows of C . (j,j) & ( for i being Element of C
for f being set st f in the Arrows of C . (i,j) holds
( the Comp of C . (i,j,j)) . (e,f) = f ) )

A4: <^o,o^> <> {} by A2;
hence e in the Arrows of C . (j,j) ; :: thesis: for i being Element of C
for f being set st f in the Arrows of C . (i,j) holds
( the Comp of C . (i,j,j)) . (e,f) = f

let i be Element of C; :: thesis: for f being set st f in the Arrows of C . (i,j) holds
( the Comp of C . (i,j,j)) . (e,f) = f

let f be set ; :: thesis: ( f in the Arrows of C . (i,j) implies ( the Comp of C . (i,j,j)) . (e,f) = f )
assume A5: f in the Arrows of C . (i,j) ; :: thesis: ( the Comp of C . (i,j,j)) . (e,f) = f
reconsider o9 = i as Object of C ;
reconsider m9 = f as Morphism of o9,o by A5;
thus ( the Comp of C . (i,j,j)) . (e,f) = m * m9 by
.= f by A3, A5 ; :: thesis: verum
end;
let i be Element of C; :: according to ALTCAT_1:def 6 :: thesis: ex b1 being set st
( b1 in the Arrows of C . (i,i) & ( for b2 being Element of the carrier of C
for b3 being set holds
( not b3 in the Arrows of C . (i,b2) or ( the Comp of C . (i,i,b2)) . (b3,b1) = b3 ) ) )

reconsider o = i as Object of C ;
consider m being Morphism of o,o such that
A6: for o9 being Object of C
for m9 being Morphism of o9,o
for m99 being Morphism of o,o9 holds
( ( <^o9,o^> <> {} implies m * m9 = m9 ) & ( <^o,o9^> <> {} implies m99 * m = m99 ) ) by A2;
take e = m; :: thesis: ( e in the Arrows of C . (i,i) & ( for b1 being Element of the carrier of C
for b2 being set holds
( not b2 in the Arrows of C . (i,b1) or ( the Comp of C . (i,i,b1)) . (b2,e) = b2 ) ) )

A7: <^o,o^> <> {} by A2;
hence e in the Arrows of C . (i,i) ; :: thesis: for b1 being Element of the carrier of C
for b2 being set holds
( not b2 in the Arrows of C . (i,b1) or ( the Comp of C . (i,i,b1)) . (b2,e) = b2 )

let j be Element of C; :: thesis: for b1 being set holds
( not b1 in the Arrows of C . (i,j) or ( the Comp of C . (i,i,j)) . (b1,e) = b1 )

let f be set ; :: thesis: ( not f in the Arrows of C . (i,j) or ( the Comp of C . (i,i,j)) . (f,e) = f )
assume A8: f in the Arrows of C . (i,j) ; :: thesis: ( the Comp of C . (i,i,j)) . (f,e) = f
reconsider o9 = j as Object of C ;
reconsider m9 = f as Morphism of o,o9 by A8;
thus ( the Comp of C . (i,i,j)) . (f,e) = m9 * m by
.= f by A6, A8 ; :: thesis: verum