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 A1, ALTCAT_1:18; :: 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 A4, A5, ALTCAT_1:def 8
.= 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 A7, A8, ALTCAT_1:def 8
.= f by A6, A8 ; :: thesis: verum