let A be QC-alphabet ; :: thesis: for p, q, r being Element of CQC-WFF A

for X being Subset of (CQC-WFF A) st X |- p => q & X |- q => r holds

X |- p => r

let p, q, r be Element of CQC-WFF A; :: thesis: for X being Subset of (CQC-WFF A) st X |- p => q & X |- q => r holds

X |- p => r

let X be Subset of (CQC-WFF A); :: thesis: ( X |- p => q & X |- q => r implies X |- p => r )

assume that

A1: X |- p => q and

A2: X |- q => r ; :: thesis: X |- p => r

X |- (p => q) => ((q => r) => (p => r)) by CQC_THE1:59;

then X |- (q => r) => (p => r) by A1, CQC_THE1:55;

hence X |- p => r by A2, CQC_THE1:55; :: thesis: verum

for X being Subset of (CQC-WFF A) st X |- p => q & X |- q => r holds

X |- p => r

let p, q, r be Element of CQC-WFF A; :: thesis: for X being Subset of (CQC-WFF A) st X |- p => q & X |- q => r holds

X |- p => r

let X be Subset of (CQC-WFF A); :: thesis: ( X |- p => q & X |- q => r implies X |- p => r )

assume that

A1: X |- p => q and

A2: X |- q => r ; :: thesis: X |- p => r

X |- (p => q) => ((q => r) => (p => r)) by CQC_THE1:59;

then X |- (q => r) => (p => r) by A1, CQC_THE1:55;

hence X |- p => r by A2, CQC_THE1:55; :: thesis: verum